










.7* A 



* .0^"^ ^^ '^.T- A 













-^^^^"^ • 











O > 






.7* A 






<U. ' 







ik^X .^-^''^'A^/^. >^\*;?a^'.V ,**'\ 



<> ' .. •' jS> 






/.c:^^% 



.o*\. 



Ao^ . 





































•• ^. 




.^^Vj^t.^V" /.• 






r.^'.^'^* 'o^r^^To^' ^0' ^- -••• 






■* 



APPLIED CALCULUS 

PRINCIPLES AND APPLICATIONS 

ESSENTIALS FOR STUDENTS 
AND ENGINEERS 



BY 

ROBERT GIBBES THOMAS 

Professor of Mathematics and Engineering at the Citadel, 
The Military College of South Carolina 



45 EXERCISES - 166 FIGURES 




NEW YORK 

D. VAN NOSTRAND COMPANY 

25 Park Place 

1919 



^ 



^% 



Copyright, 1919, 

BY 

D. VAN NOSTRAND COMPANY 



NOV -5 1819 



Stanbopc iptcss 

F. H.GILSON COMPANY 
BOSTON, U.S. A 



©CI.A5';!6 4a8 



PREFACE 



This book as a first course in the Calculus is not designed 
to be a complete exposition of the Calculus in either its 
principles or its applications. It is an effort to make clear 
the basic principles and to show that fundamental ideas are 
involved in famihar problems. While formulas and alge- 
braic methods are necessary aids to concise and formal 
presentation, they are not essential to the expression of the 
principles and imderlying ideas of the Calculus. These can 
be expressed in plain language without the use of symbols 
— one writer challenging the citing of a single instance 
where it cannot be done. 

The practice is common, at least with ''thoughtless think- 
ers," of bUndly using formulas without any true conception 
of the ideas for which they are but the symbolic expres- 
sion. The formulas of the Calculus are an invaluable aid in 
economy of thought, but their effective use is dependent 
upon an adequate knowledge of their derivation. The 
object of this book is to set forth the methods of the Cal- 
culus in such a way as to lead to a working and fruitful 
knowledge of its elements, to exhibit something of its power, 
and to induce its use as an efficient tool. No claim is made 
for absolute rigor in all the deductions, but confidence is 
invited in the soundness of the reasoning employed and in 
the logical conclusions obtained. 

There are students, and engineers also, who when con- 
strained to use the Calculus look upon it as a necessary evil. 
This attitude is without doubt due to their minds having 
never had a firm grasp upon its principles nor a full reahza- 



IV PREFACE 

tion of the efficiency of its methods. A student while tak- 
ing a course in the Calculus usually spends at least one-half 
his effort in reviewing previous mathematics. In fact, a 
course in the Calculus is held to involve an excellent review 
of geometry, algebra, and especially of trigonometry; hence, 
at the end of a term it is too much to expect of the average 
student that he have an adequate knowledge of the Calculus. 

If a choice must be made between the ability to solve 
equations (including integration processes) and the far more 
rare ability to set up equations to represent established facts 
and laws, there can be little question as to which type of 
abihty should be cultivated. The latter is of higher order 
and is hkely to include the former. Engineers, physicists, 
inventors and men of science generally find it difficult to 
translate their observations into language which the pure 
mathematician can understand. In fact, such translation 
usually involves the writing of the equation : an undertaking 
beyond the capacity equally of the non-mathematical scientist 
and the pure mathematician. Integration of the equation, 
once set up, the mathematician will undertake; conceiv- 
ably, so might a machine. Fruitful deductions and rules 
of practice result. The difficulty of reahzing these results 
arises not from difficulties in moving about the symbols, but 
from inabihty on the part of nearly all persons to state facts 
in terms of symbols. It is as if no harmonist knew a melody 
and no melodist knew a note. This book aims to keep fact 
and symbol in close association, so that the student will 
never use the latter without being conscious of the former. 
It may then be expected that he will ultimately be able to 
visualize the symbolic expression when the fact is known. 

Apart from the references in the text and in footnotes, 
acknowledgment is here made of the clarifying and logical 
ideas embodied in the books on the Calculus by Gibson, by 
Taylor, and by Townsend and Goodenough; also in Hedrick's 
paper on the Calculus without Symbols. 



PREFACE V 

The introduction to this book ends with a reference to 
the discoverer of the Calculus. It is deemed not unfitting 
that the b6ok should close with the Central Forces of the 
Principia. 

ROBEKT GiBBES ThOMAS. 

The Citadel, Charleston, S. C. 
February 1st, 1919. 



CONTENTS 



INTRODUCTION. 
PART I. DIFFERENTIAL CALCULUS. 



CHAPTER I. 

FUNCTIONS. DIFFERENTIALS. RATES. 

Abticle Page 

1. Variables and Constants 5 

2. Functions. Dependent and Independent Variables 6 

3. Function — Continuous or Discontinuous 7 

4. Representation of Functional Relation 7 

5. Function — Increasing or Decreasing 8 

6. Classes of Functions. Empirical Equations 9 

7. Increments 12 

Exercise I 13 

8. Uniform and Non-uniform Change 14 

9. Differentials 15 

10. Illustrations of Differentials 15 

11. Rate, Slope, and Velocity 18 

12. Rate, Speed, and Acceleration 19 

13. Rate and Flexion 21 

14. Illustrations 21 

Exercise II 23 

CHAPTER II. 
DIFFERENTIATION. DERIVATIVES. LIMITS. 

15. Derivative 24 

16. Differentiation 25 

17. Limits 25 

18. Theorems of Limits 26 

19. Derivative as a Limit. Function of a Function 26 

20. Illustrative Examples 31 

21. Replacement Theorem 37 

22. Limit of Infinitesimal Arc and Chord 37 

vii 



viii CONTENTS 

ALGEBRAIC FUNCTIONS. 

Article Page 

23. Formulas and Rules for Differentiation 38 

24-31. Derivation of Formulas 39-43 

Exercise III 45 

LOGARITHMIC AND EXPONENTIAL FUNCTIONS. 

32. Formulas and Rules for Differentiation 47 

33. Derivation of Formulas 48 

34. Limit f 1 + IV = e 50 

35. Derivation of Formulas 52 

36. Limit f 1 + -V = e== 52 

37. Derivation of Formulas. 53 

38. Modulus 54 

39. Logarithmic Differentiation 56 

40. Relative Rate. Percentage Rate 56 

Exercise IV 57 

41. Relative Error 59 

42. Compound Interest Law 60 

TRIGONOMETRIC FUNCTIONS. 

43. Circular or Radian Measure 64 

44. Formulas and Rules for Differentiation 64 

45. Derivation of Formulas 65 

46. Limit , . - i ^ nr> 

1 bo 



m 



47-50. Derivation of Formulas 68 

51. Note on Formula 69 

52. Remarks on Formula '70 

Exercise V 71 

53. The Sine Curve or Wave Curve 72 

54. Damped Vibrations 73 

INVERSE TRIGONOMETRIC FUNCTIONS. 

55. Formulas and Rules for Differentiation 76 

56-59. Derivation of Formulas 77, 78 

Exercise VI 78 



CONTENTS ix 

Article Page 

60. Hyperbolic Functions 80 

61. General Relations 80 

62. Numerical Values. Graphs 80 

63. Derivatives 81 

64. The Catenary 81 

65. Inverse Functions 81 

66. Derivatives of Inverse Functions 82 

CHAPTER III. 

SUCCESSIVE DIFFERENTIATION. ACCELERATION. CURVI- 
LINEAR MOTION. 

67. Successive Differentials x 84 

68. Successive Derivatives 84 

69. Resolution of Acceleration 86 

Exercise VII 87 

70. Circular Motion 88 

71. The Second Law of Motion 90 

72. Angular Velocity and Acceleration 92 

73. Simple Harmonic Motion 94 

74. Self-registering Tide Guage 97 

Exercise VIII : . 97 



CHAPTER IV. 
GEOMETRICAL AND MECHANICAL APPLICATIONS. 

75. (a) Tangents and Normals 99 

(6) Subtangents and Subnormals 100 

76. Illustrative Examples 101 

Exercise IX 107 

77. Polar Subtangent, Subnormal, Tangent, Normal 108 

Exercise X 110 

CHAPTER V. 

MAXIMA AND MINIMA. INFLEXION POINTS. « 

78. Maxima and Minima Ill 

79. The Condition for a Maximum or a Minimum Value.. . Ill 

80. Graphical Illustration 113 

81. Rule for Applying Fundamental Test 115 



X CONTENTS 

ArticixE Page 

82. Rule for Determining Maxima and Minima 115 

83. Inclusive Rule 116 

84. Typical Illustrations 118 

85. Inflexion Points 120 

86. Polar Curves 123 

87. Auxiliary Theorems 124 

Exercise XI 125 

Problems in Maxima and Minima 127 

Determination of Points of Inflexion 132 

CHAPTER VI. 
CURVATURE. EVOLUTES. 

88. Curvature 133 

89. Curvature of a Circle 134 

90. Circle, Radius, and Center of Curvature 135 

91. Radius of Curvature in Rectangular Coordinates 137 

92. Approximate Formula for Radius of Curvature 138 

Exercise XII 139 

93. Radius of Curvature in Polar Coordinates 140 

Exercise XIII 142 

94. Coordinates of Center of Curvature 142 

95. Evolutes and Involutes 143 

96. Properties of the Involute and Evolute 143 

97. To find the Equation of the Evolute 144 

CHAPTER VII. 

CHANGE OF THE INDEPENDENT VARIABLE. FUNCTIONS 
OF TWO OR MORE VARIABLES. 

98. Different Forms of Successive Derivatives 149 

99. Change of the Independent Variable 149 

Exercise XIV 151 

100. Function of Several Variables 152 

101. Partial Differentials 152 

102. Partial Derivatives 153 

103. Tangent Plane. Angles with Coordinate Planes 154 

Exercise XV 156 

104. Total Differentials 156 

105. Derivative of an Implicit Function 158 

Exercise XVI 158 



CONTENTS XI 

Abticle Page 

106. Total Derivatives 159 

107. Illustrative Examples 160 

Exercise XVII 161 

108. Approximate Relative Rates and Errors 162 

Exercise XVIII 163 

109. Partial Differentials and Derivatives of Higher Orders . 164 

110. Interchange of Order of Differentiation 165 

Exercise XIX 166 

111. Exact Differentials 167 

Exercise XX 170 

112. Exact Differential Equations 170 



PART n. INTEGRAL CALCULUS 



CHAPTER I. 
INTEGRATION. STANDARD FORMS. 

113. Inverse of Differentiation 171 

114. Indefinite Integral 173 

115. Illustrative Examples 174 

116. Elementary Principles 178 

117. Standard Forms and Formulas 180 

118. Use of Standard Formulas 182 

Exercise XXI 185 

Exercise XXII 187 

119-121. Derivation of Formulas 187-190 

Exercise XXIII 191 

122. Reduction Formulas. 194 

123. Integration by Parts '. . . 194 

Exercise XXIV " 197 

124. Reduction Formulas for Binomials 198 

Exercise XXV 199 

CHAPTER II. 
DEFINITE INTEGRALS. AREAS. 

125. Geometric Meaning of Cf (x) dx 204 

126. Derivative of an Area 205 

127. The Area under a Curve 206 



XU CONTENTS 

Article Page 

128. Definite Integral 206 

129. Positive or Negative Areas 208 

130. Finite or Infinite Areas — " Limits " Infinite 208 

131. Interchange of Limits 210 

132. Separation into Parts 210 

133.- Mean Value of a Function 211 

134. Evaluation of Definite Integrals 213 

Exercise XXVI 213 

135. Areas of Curves 214 

Exercise XXVII 216 

136. To find an Integral from an Area 219 

137. Area under Equilateral Hyperbola 221 

138. Significance of Area as an Integral 223 

139. Areas imder Derived Curves 225 

CHAPTER III. 

INTEGRAL CURVES. LENGTHS OF CURVES. CURVE OF A 
FLEXIBLE CORD. 

140. Integral Cm-ves 227 

141. Application to Beams 229 

142. Lengths of Curves 236 

Exercise XXVIII 237 

143. Lengths of Polar Curves 239 

Exercise XXIX 240 

144. Curve of a Cord under Uniform Horizontal Load — 

Parabola 241 

145. The Suspension Bridge 244 

146. Curve of a Flexible Cord — Catenary 245 

147. Expansion of cosh x/a and sinh x/a 248 

148. Approximate Formulas 249 

149. Solution of s = a sinh x/a. . 250 

150. The Tractrix 254 

151. Evolute of the Tractrix 256 

CHAPTER IV. 

INTEGRATION AS THE LIMIT OF A SUM. SURFACES AND 

VOLUMES. 

152. Limit of a Sum 259 

153. The Summation Process , 261 



CONTENTS Xlll 

Article Page 

154. Approximate and Exact Summations 262 

Exercise XXX 266 

155. Volumes 267 

156. Representation of a Volume by an Area 268 

157. Surface and Volume of any Frustum 270 

Exercise XXXI.. 282 

158. Prismoid Formula 283 

159. Application of the Prismoid Formula 284 

Exercise XXXII 287 

160. Surfaces and Solids of Revolution 288 

Exercise XXXIII 292 



CHAPTER V. 

SUCCESSIVE INTEGRATION. MULTIPLE INTEGRALS. 
SURFACES AND VOLUMES. 

161. Successive Integration 296 

Exercise XXXIV 299 

162. Successive Integration with Respect to Two or More 

Independent Variables 300 

163. The Constant of Integration 301 

Exercise XXXV 303 

164. Plane Areas by Double Integration — Rectangular Co- 

ordinates 304 

Exercise XXXVI 307 

165. Plane Areas by Double Integration — Polar Coordi- 

nates 308 

Exercise XXXVII 311 

166. Area of any Surface by Double Integration 311 

Exercise XXXVIII 316 

167. Volumes by Triple Integration — Rectangular Coor- 

dinates 317 

Exercise XXXIX 320 

168. Solids by Revolution by Double Integration 321 

169. Volumes by Triple Integration — Polar Coordinates. . . 321 

170. Volumes by Double Integration — Cylindrical Coor- 

dinates 324 

171. Mass. Mean Density 327 



XIV CONTENTS 

CHAPTER VI. 
MOMENT OF INERTIA. CENTER OF GRAVITY. 

Article Page 

172. Moment of a Force about an Axis 331 

173. First Moments 331 

174. Center of Gravity of a Body 332 

175. Center of Gravity of a Plane Surface 333 

176. Center of Gravity of any Surface 334 

177. Center of Gravity of a Line 334 

178. Center of Gravity of a System of Bodies 335 

179. The Theorems of Pappus and Guldin 336 

Exercise XL 342 

180. Second Moments — Moment of Inertia 344 

181. Radius of Gyration 345 

182. Polar Moment of Inertia 345 

183. Moments of Inertia about Parallel Axes 346 

184. Product of Inertia of a Plane Area 348 

185. Least Moment of Inertia 349 

186. Deduction of Formulas for Moment of Inertia 350 

187. Moment of Inertia of Compound Areas 352 

CHAPTER VII. 
APPLICATIONS. PRESSURE. STRESS. ATTRACTION. 

188. Intensity of a Distributed Force 355 

189. Pressure of Liquids 356 

Exercise XLI 362 

190. Attraction. Law of Gravitation 363 

191. Value of the Constant of Gravitation 375 

192. Value of the Gravitation Unit of Mass 376 

193. Vertical Motion under the Attraction of the Earth .... 376 

194. Necessary Limit to the Height of the Atmosphere 378 

195. Motion in Resisting Medium 379 

196. Motion of a Projectile 380 

197. Motion of Projectile in Resisting Medium 383 

CHAPTER VIII. 
INFINITE SERIES. INTEGRATION BY SERIES. 

198. Infinite Series 385 

199. Power Series 386 



CONTENTS XV 

Article Page 

200. Absolutely Convergent Series 388 

201. Tests for Convergency 390 

Exercise XLII 392 

202. Convergency of Power Series 393 

203. Integration and Differentiation of Series 394 



CHAPTER IX. 

TAYLOR'S THEOREM. EXPANSION OF FUNCTIONS. 
INDETERMINATE FORMS. 

204. Law of the Mean ^ . . 401 

205. Other Forms of the Law of the Mean 403 

206. Extended Law of the Mean 405 

207. Taylor's Theorem 405 

208. Another Form of Taylor's Theorem 406 

209. Maclaurin's Theorem 407 

210. Expansion of Functions in Series 408 

211. Another Method of Deriving Taylor's and Maclaurin's 

Series 408 

212-214. Expansion by Maclaurin's and Taylor's Theorems 411 

215. Examples 412 

Exercise XLIII 418 

216. The Binomial Theorem 419 

217. Approximation Formulas 421 

Exercise XLIV 424 

218. Application of Taylor's Theorem to Maxima and 

Minima 424 

219. Indeterminate Forms 425 

220. Evaluation of Indeterminate Forms 428 

221. Method of the Calculus 430 

Exercise XLV 434 

222. Evaluation of Derivatives of Implicit Functions 435 



CHAPTER X. 

DIFFERENTIAL EQUATIONS. APPLICATIONS. CENTRAL 
FORCES. 

223. Differential Equations 436 

224. Solution of Differential Equations 436 

225. Complete Integral 437 



XVI CONTENTS 

Article Page 

226. The Need and Fruitfulness of the Solution of Differ- 

ential Equations 441 

227. Equations of the Form Mdx + Ndy = 444 

228. Variables Separable 446 

229. Equations Homogeneous in x and y 446 

230. Linear Equations of the First Order 448 

231. Equations of the First Order and nth Degree 449 

232. Equations of Orders above the First 449 

233. Linear Equations of the Second Order 453 

APPLICATIONS. 

234. Rectilinear Motion 456 

235. Curvilinear Motion , . . . 460 

236. Simple Circular Pendulum 461 

237. Cycloidal Pendulum 464 

238. The Centrifugal Railway 466 

239. Path of a Liquid Jet 467 

240. Discharge from an Orifice 469 

CENTRAL FORCES. 

241. Definitions 472 

242. Force Variable and Not in the Direction of Motion . . . 472 

243. Kepler's Laws of Planetary Motion 477 

244. Nature of the Force which Acts upon the Planets 478 

245. Newton's Verification 480 

Index 483-490 



What we call objective reality is, in the last analysis, what is common 
to many thinking beings and could be common to all; this common 
part . . . can only be the harmony expressed by mathematical laws. 

— H. POINCARE 



APPLIED CALCULUS. 



INTRODUCTION. 

The Calculus treats of the rates of change of related 
variables. The factors of life are ever changing, acting and 
reacting upon each other. The quantities with which we 
have to deal in ordinary affairs are for the most part in a 
state of change. Hence the field in which the principles of 
the Calculus are directly involved is a wide one. 

In observing the changes about us we note that they take 
place at various rates, and the determination of the rapidity 
of the change may be the controlling factor in many investi- 
gations. Whenever the rapidity of the change of anything 
is in question, the methods of the Calculus have appropriate 
application. 

In the case of velocity or speed, there is rate of change of 
distance and time ; in a thermometer we have rate of change 
of length and temperature, while in the barometer there is 
rate of change of height and density; in the slope of ground 
or grade of a road we have rate of change of vertical height 
and horizontal distance; and in the case of a curve, rate of 
change of ordinate and abscissa, or slope of the curve. 

In the case of a body in variable motion, it becomes 
desirable to determine its velocity at some point of its path 
or at some instant of time, that is, the instantaneous velocity. 
This notion of rate of change at an instant is common even 
to untrained minds. 

When one says of a train in variable motion, that it is 
now going at the rate of sixty miles an hour, one means that 

1 



2 APPLIED CALCULUS 

at the instant considered the rate is such that, if it were 
maintained, the train would go sixty miles in an hour, that 
being the instantaneous velocity. 

The method of the Calculus in getting the rate of change 
of a variable at any instant is in accordance with natural 
procedure: measure the amount of change in a short period 
of time, then the average rate of change during that period 
is the ratio of amount of change to length of period; the 
limit approached by this ratio, as the period of time is 
diminished towards zero as a limit, is the rate of change at 
the instant the period began. 

In determining the greatest and least values of a variable 
quantity, they are found where the rate of change of the 
variable is zero. For instance, at the maximum and mini- 
mum temperature during a day, the rate of change of the 
temperature is zero. There is a difference, however, in that 
before the hottest moment the temperature was rising and 
then afterwards falling, while at the coldest moment the 
reverse was the case. In both cases the temperature's rate 
of change was momentarily zero. Here is to be seen the 
method of the Calculus as to maxima and mimina. 

A distinguishing feature of the Calculus is that in addition 
to real sensible quantities it uses ideal hypothetical concepts, 
which are quantities that exist if certain conditions are 
maintained. The Calculus connects these two classes of 
quantities. Passing from the real to the ideal is Differentia- 
tion, from the ideal to the real is Integration. The advantage 
of introducing ideal quantities is that in many problems an 
expression for the ideal is readily formed and from this 
expression the real quantity is obtained by Integration. In 
other cases the real quantity being given, the problem is 
solved by the ideal quantity, obtained from the real by 
Differentiation. 

The might of the invisible and intangible forces in Nature, 
predicated upon a concept {the aether), is generally recognized 



INTRODUCTION 3 

in this day and generation. Therefore, it is not to be 
wondered at, that in deahng with material things and in 
seeking the inner law by which they act and react upon each 
other, we should call to our aid ideal concepts. The exclu- 
sive realist in his passion for facts is prone to overlook the 
fact that ideas are the first of facts. 

It is acknowledged that science is useless unless it teaches 
us something about reahty. Let it be acknowledged that 
the aim of science is not things themselves, but the relations 
between things, and the fruitfulness of the ideal quantities 
of the Calculus is recognized. The differentials employed, 
when properly defined, are not '^ ghosts of departed quan- 
tities," even if in some cases ideal in character. "A^r2/," 
perhaps, but never ^' nothing,'^ they give to the creation of 
the mind "a local habitation and a name J' 

While the ratio of some real quantities may never equal 
the ratio of the ideal quantities, nevertheless the former 
ratio may approach so closely the latter as a limit that the 
exact value of the ideal ratio can be discerned. So the 
differentiation of any real variable quantity is possible. On 
the other hand, the exact integration of every ideal quantity 
is not possible, for in some cases no corresponding real 
quantity exists. 

In this respect there is an analogy with Involution and 
Evolution. Any number may be raised to a power; but 
the exact root of every number cannot be found, for no real 
root exists for some numbers. 

Differential Calculus deals with the rates of change of 
continuous variables when the relation of the variables is 
known. 

Integral Calculus is concerned with the inverse problem of 
finding the relation of the variables themselves when their 
relative rates are known. 

While some problems to which the Calculus is applied may 
be solved by other methods, it often furnishes the sunplest 



4 APPLIED CALCULUS 

solution; and there are cases in which the Calculus alone 
gives the solution. The Calculus is a tool for the efficient 
worker, and in the hands of skillful investigators the Cal- 
culus has proved to be a powerful instrument in bringing to 
Ught the truths of Nature. 

In reference to the mighty intellect that conceived it, there 
is pardonable hyperbole in the lines of the Poet : — 

" Nature and nature's laws lay hid from sight, 
God said, 'Let Newton he,' and all was 



ERRORS AND OMISSIONS 

Page 24, on last line, f{x) = m for f(x) = m. 
161, in Example 3, 35 for 36. 
174, on figure, X misplaced. 
196, on second line, integral sign omitted. 
212, on fifth line, CPi for CP. 
232, on fifth line, integral sign omitted. 
251, Example* 1, *Miller and Lilly's Analytic Mechanics. 
253, at bottom of page, True should be omitted. 
259, on second line. Art. 124 for Art. 128. 
263, at end of fourth line, period for comma. 
345, in equation, x^ for y"^. 
358, in (2), x for x. 
385, in expansion, factor a omitted. 

418, in Note, -w for x in (2) gives e^'^ = 1, should be, -2ir for x in 
(2) or 27r for x in (1) gives e^^'^ = 1, whence e^'^ = ±1 
hence ... 
456, reference, (Ex. 6, Art. 116) for (Ex. 6, Art. 115). 



PART I. 

DIFFERENTIAL CALCULUS. 



CHAPTER I. 
FUNCTIONS. DIFFERENTIALS. RATES. 

1. Variables and Constants. — A variable is a quantity 
that changes in value. It is said to vary continuously when, 
in changing from one value to another, it takes each inter- 
mediate value successively and only once. If at any value 
it ceases to vary continuously, it is said to be discontinuous 
at that value. 

A constant is a quantity whose value is fixed. If its value 
is always the same in every discussion, it is an absolute 
constant. If the fixed value may be different in different 
discussions, it is an arbitrary constant. 

In the equation of the circle, x^ -{- y^ = r^, x and y are 
variables that vary continuously from to d= r ; while r is an 
arbitrary constant, as its value is fixed only for any one circle. 

In the ordinary affairs of life we have to deal with con- 
tinuous variables, such as time; the distance of an object in 
continuous motion from any point on its path; and with 
discontinuous variables, such as the amount of a sum of 
money at interest compounded periodically; the price of 
cotton; the cost of money orders, etc. 

In nature we have constants, such as: the mass of a body, 
which is an absolute constant; the weight of a body, which 
is an arbitrary constant, as it is fixed according to latitude 

5 



6 DIFFERENTIAL CALCULUS 

and elevation; in mathematics, the ratio of the circum- 
ference of a circle to its diameter and the base of Naperian 
logarithms are absolute constants. ^ 

Variables are represented usually by the last letters of the 
alphabet; as x, y, z, or p, 6, 0, etc. The letter A, however, 
often represents a variable area. ^ 

Absolute constants are denoted by number symbols, and 
there are some absolute constants represented by letters, as 
TT, e, for the ratio and base just mentioned, each transcen- 
dental but the most important in mathematics. 

Arbitrary constants are represented usually by the first 
letters of the alphabet; as a, b, c, a, /3, y, etc. Particular 
values of variables are constants and are denoted by Xi, 2/1, ; 
Zu X2, 2/2, ^2, etc. 

2. Functions. Dependent and Independent Vari- 
ables. — When two variables are so related that the value 
of one of them depends upon the value of the other, the first 
is the dependent variable and is said to be a function of the 
second, and the second is the independent variable, which in 
connection with the function is usually called simply the 
variable, or sometimes the argument. 

The area of a square is a function of the length of a side. 
The area or the circumference of a circle is a function of its 
radius. The square, or the square root, or the logarithm of 
a number, is a function of the number. 

Any function of x is represented by / (x), F (x), <l) (x), etc., 
and the symbol f{x) denotes any expression involving x, 
whose value depends upon the value oi x. In any discussion 
involving x, f (a) means the value of / (x) when x is replaced 
by a throughout the expression. In y =f{x), x is the 
independent variable and y is the function or dependent 
variable. In the equation x'^ -\- y^ = r^, y = Vr^ — x^ or 
X = Vr^ — y^, so y == f (x) or x = f (y). If one variable is 
expressed directly in terms of another, the first is said to be 
an explicit function of the second. If the relation between 



REPRESENTATION OF FUNCTIONAL RELATION 7 

the two variables is given by an equation containing them 
but not solved for either, then either variable is said to be 
an implicit function of the other. So in x"^ -\- y^ = r^, y is 
an implicit function of x and x is an implicit function of y; 
but in y = v r^ — x"^, y is an exphcit function of x, and in 
X = Vr^ — ?/2, X is an explicit function of y. 

A variable may be a function of more than one variable, 
thus in ^2 = x^ + y^, or in 2; = xy, 2; is a function of x and y. 
The area of a rectangle is a function of its base*and altitude. 
The volume of a solid is a function of its three dimensions; 
so in y = xyz, V = f (x, y, z). 

3. Function — Continuous or Discontinuous. — A func- 
tion as/ (x) is said to be continuous between x = a and x = h, 
if when x varies continuously from a to h, f {op) varies con- 
tinuously from /(a) to f{h). In other words, f (x) is con- 
tinuous between x = a and x = b when the locus oi y = 
fix) between the points (a, /(a)) and (b, f (h)) is an un- 
broken line, straight or curved. 

A function is said to be discontinuous at any value when 
it ceases to vary continuously at that value, even though its 
variable may be continuous. 

Some functions are continuous for all values of their 
variables; others are continuous only between certain 
limits. For example, sin d and cos d are continuous for 
values of 6 from 6 = to ^ = 2 tt ; tan 9 is continuous from 
= to ^ = 7r/2 and from d = t/2 lo 6 = '^ it, but when 
6 passes through 7r/2 or | tt, tan changes from +00 to 
— GO , hence tan 6 is discontinuous for 6 = 7r/2 or | tt. 

The Calculus treats of continuous variables and functions 
only, or of variables and functions between their limits of 
continuity. 

4. Representation of Functional Relation. — Often the 
relation between the function and the argument can be 
expressed by a simple formula. For example, if s is the 
distance fallen from rest in time t, then s = / (0 = J gt^. 



8 DIFFERENTIAL CALCULUS 

In such cases, the value of the function for any value 
taken for the variable can be found by simply substituting 
in the formula; thus, 

si=f{l) =ig, S2=f(2) = ig.2' = 2g, 
and so for any value. 

A function is tabulated when values of the argument, as 
many and as near together as desired, are set down in one 
column and the corresponding values of the function are set 
down opposite in another column. For example, in a table 
of sines, the angle in degrees and minutes is the argument, 
and the sine of the angle is the function. 

A function is graphed or exhibited graphically by laying 
off the values of the argument as abscissas along a horizontal 
axis, and at the end of each abscissa erecting an ordinate 
whose length will represent the corresponding value of the 
function; a curve drawn through the tops (or bottoms) of 
the ordinates is called the curve, or the graph of the function. 

If y = f(x), the curve is the locus of the equation; but 
it is the length of the ordinate up (or down) to the curve, 
rather than the curve itself, that represents the function. 

If p = f (6), the function may be graphed by laying off at 
a point on a line as axis the various angles, — values of the 
argument 6, and along the terminal sides of the angles the 
corresponding values of the function p; a curve through the 
ends of the vectorial radii will be the graph of the function, 
and will be a polar curve. Here, too, it is the length of the 
radius to the curve, rather than the curve itself, that repre- 
sents the function. The area under a curve may be taken 
to represent a function while the ordinate or radius repre- 
sents some other function. (See Art. 139.) 

5. Function — Increasing or Decreasing. — An increas- 
ing function is one that increases when its variable increases, 
hence, it decreases when its variable decreases. A decreasing 
function is one that decreases when its variable mcreases, 
hence it increases when its variable decreases. Thus ax and 



CLASSES OF FUNCTIONS 9 

a^ are increasing, and ajx and a — ic are decreasing functions 
of X. 

6. Classes of Functions. — An algebraic function is one 
that without the use of infinite series can be expressed by the 
operations of addition, subtraction, multipHcation, division 
and the operations denoted by constant exponents. 

The common forms are: (a =b bx), (a =b bx""), ax, a/x, x"^, 
including x'^, x^, Vx, 1/Vx, etc. 

All functions which are not algebraic are called trans- 
cendental. Of these, the most important are : 

The exponential functions, y = a^ or 6^, and y = e^, and 
their inverse forms, the logarithmic functions, 
X = loga y or log6 y and x = log^ y. 

The trigonometric functions, y = smd,x = cos d,y = tan 6, 
and the inverse trigonometric functions, B = arc sin y or sin~^ y, 
6 = arc cos x or cos~^ x, 6 = arc tan y or tan"^ y. 

The hyperbolic functions, sinh a; = (e^ — e~^)/2, cosh a; = 
(e^ + e"^)/2, tanho: = (e^ — e~=')/e^ + e"^); and the inverse 
hyperbolic functions, 

sinh~i X = y = log {x + Va:^ + l), 
cosh"! X = y = \og{x± Vx'^ — l), 
tanh~i a; = 2/ = i log (1 + x/1 — x). 

Note. — The phenomena of change in Nature, in general, 
have been found to be in accordance with one or the other 
of three fundamental laws. These have been stated * to be 
the parabolic law, expressed by the power function y = ax^'j 
where n is constant, positive or negative; the harmonic law, 
expressed by the periodic function y = a sin (mx) ; and the 
law of organic growth, or the compound interest law, expressed 
by the exponential function y = ae^^. It is to be noted that 
as X increases in arithmetic progression, y oi the exponential 
function increases in geometrical progression; while, as x 

* In Elementary Mathematical Analysis, by Charles S. Slichter. 



10 DIFFERENTIAL CALCULUS 

increases in geometrical progression, y of the power function 
increases in geometrical progression also. 

Examples in Nature of the working of these three laws 
will be given later. 

EMPIRICAL EQUATIONS. 

Very often the form of a function is given only empirically; 
that is, the values of the function for certain values of the 
variable are known from experiment or observation, and the 
intermediate values are not given; for example, the height 
of the tide read from a gauge every hour. 

In such cases the Calculus is not of much use unless some 
known mathematical law can be found which represents the 
function sufficiently accurately. 

This "problem of finding a mathematical function whose 
graph shall pass through a series of empirically given points is 
of great practical importance. 

The known values of the function and of the variable are 
plotted on cross-section paper, 'logarithmic squared paper" 
greatly facilitating the solution, and a smooth curve being 
drawn 'Ho fit" the determined points, the equation of this 
curve is required. The curve suggested by the plotted 
points may have for its equation one of the following forms : 

(straight line) y = a -\-bx, or y = mx; 

(parabola) . y = a -{-bx -\- cx^, ov y = a -{- cx^) 

(hyperbola) y = a -\- c/x -{-b, y = 1/x", or 

xy = bx-\- ay; 
(sine curve) y = ci sin (bx -\- c), or y = a sin {mx) ; 

(power function) y = ax" (n any number) ; 

(exponential function) y = ae^^. 

If the curve suggested by the plotted poirits is a straight 
line, determine the values of a and b, or of m, from the 
observed data. The straight line is not likely to pass 
through all the points plotted, even when the straight-line 



EMPIRICAL EQUATIONS 11 

law is the correct expression of the relation to be determined; 
for the experimental data are subject to error. If the line 
fits the points within the limits of accuracy of the experiment, 
it may be drawn through two of the plotted points, and a 
and 6, or m, may be evaluated from their coordinates. 

By appropriate treatment of the data many of the laws 
can be transformed into a linear relation. Thus, when the 
points plotted suggest a vertical parabola with its vertex on 
the 2/-axis, the required equation will be of the form, 

y = a -\- cx^. (1) 

If t is put for x^ in (1), and the values of t and y plotted, 
these values satisfy the relation y = a -\- ct, that is, a straight- 
line law. The power function y = ax"^ may be expressed : 

log y = log a -\- n log x, (2) 

that is, the logarithms of the given data satisfy a straight-line 
law. The straight-line law to fit the logarithms can be 
determined and compared with (2) to find a and n, which 
are substituted in i/ = ax'^. 

The hyperbolic law and the exponential function also can 
be transformed to the straight-fine law, and the constants 
evaluated. Whether the experimental data can be expressed 
by a power function or by an exponential function can be 
determined by a test. When the data show that, as the 
argument changes by a constant factor, the function also 
changes by a constant factor, then, the relation can be 
expressed by a power function. 

When, however, it is found that a change of the argument 
by a constant increment changes the function by a constant 
factor, then the relation can be expressed by an equation of 
the exponential type. (See Note, Art. 6.) 

A full discussion of this problem of finding the expression 
of the relation between a function and its argument from 
limited experimental data involves the theory of least 
squares, and is out of place in a first course in the Calculus. 



12 DIFFERENTIAL CALCULUS 

This necessarily inadequate treatment of the subject here is 
warranted by the importance of the problem. 

"^Example. — The amount of water Q, in cu. ft. that flows 
through 100 feet of pipe of diameter d, in inches, with initial 
pressure of 50 lbs. per sq. in. is given by the following: 

(i 1 1.5 2 3 4 6 

Q 4.88 13.43 27.50 75.13 152.51 409.54 

Find a relation between Q and d. 

Let X = log d, y = log Q ; then the values of x and y are : 

x = \ogd 0.000 0.176 0.301 0.477 0.602 0.788 
y = logQ 0.688 1.128 1.439 1.876 2.183 2.612 

These values plotted give points in the xy plane very nearly 
on a straight line ; therefore, taking y = a -\- bx, a and h can 
be evaluated by measurement on the figure; 

a = 0.688 = log 4.88, h = 2.473. 

Hence, log Q = log 4.88 + 2.473 log d = log (4.88 d'-^'^) ; 

whence Q = 4.88 d^-^^s. * (Ziwet and Hopkins.) 

7. Increments. — The amount of change in the value of 
a variable is called an increment. If the variable is increas- 
ing, its increment is positive ; if it is decreasing, its increment 
is negative and is really a decrement. 

An increment of a variable is denoted by putting the letter 
A before it; thus Ax, A.y and A{x'^) denote the increments 
of X, y, and x^, respectively. If y = fix), Ax and A?/ denote 
corresponding increments of x and y, and 

^y = ^f{x)=J{x^-^x)-f{x), ■ 

^.^ Aj/_ A/(x) ^ /(x + Ax)-/(x) ^ 
" Ax Ax Ax 

X denoting any value of x. 

In the figure, let OPi ... S be the locus oi y = f (x) 
referred to the rectangular axes OX and OY. If when x = 



INCREMENTS 



13 



OMi, Ax = M1M2, then Ay = M2P2 - MiPi = DP2; if 
when X =OMz, Ax = M3M4, then Ay = MJ'^-M^P^ = -EP^. 




In the last case Ay is negative and is what algebraically 
added to M3P3 gives M4P4. When 

X = OMi = xi, / (x) = MiPi = / (xi) ; 

when 

X = OM2 = xi + Ax, /(x) = M2P2 = /(xi + Ax); 
hence when 

X = xi, A/ (x) = M2P2 - MiPi = / (xi + Ax) - / (xi). 



, EXERCISE I. 

1. One side of a rectangle is 10 feet. Express the variable area A 
as a function of the other side x. 

2. Express the circumference of a circle as a function of its radius 
r; of its diameter d. 

3. Express the area of a circle as a function of its radius r; of its 
diameter d. 

4. Express the diagonal d of a square as a function of a side x. 

5. The base of a triangle is 10 feet. Express the variable area A 
as a function of the altitude y. 

6. If y =f{x), y -^^y =f{x + ^x); 

:. Ay = A/ (x) =f{x+Ax)-f (x), 

and hence, 

Ay ^ Af(x) _ f(x+Ax) - f (x) 

Ax Ax Ax 

All 
If y = mx + 6, find value of Ay and of --^« 

Ax 



7. If ?/ = x2, find value of Aw and of ^ 



14 DIFFERENTIAL CALCULUS 

^. 
Ax 

Aw 

8. If w = x^, find value of Aw and of -—' 

^ ^ ^ Ax 

9. li y = x^, find value of A^/ and of --^, assuming the binomial 
theorem. 

10. li y = f {x) = mx -\- b, write values of 



/(O), /(I), /(-I), /(^)- 



8. Uniform and Non-uniform Change. — When the ratio 
of the corresponding increments of two variables is constant, 
either variable is said to change uniformly with respect to 
the other. 

When y = mx + h, ■—- = m (constant). (6, Exercise I.) 

It follows that any linear function of x changes uni- 
formly with respect to x; that is, y changes uniformly with 
respect to x when the point (x, y) moves along any straight 
line. 

When the ratio of the corresponding increments of two 

variables is variable, either variable is said to change non- 

uniformly with respect to the other. 

Av 
When 2/ = a;^, -r-^ = 2 X + Ax (variable). (7, Exercise I.) 

Thus the area of a square changes non-uniformly with 
respect to a side. Any non -linear function of x changes 
non-uniformly with respect to x, for evidently y changes non- 
uniformly with respect to x when the point {x, y) moves 
along any curved line. 

Since time changes uniformly, any variable will change 
uniformly when it receives equal increments in equal times; 
and it will change non-uniformly when it receives unequal 
increments in equal times. 

Thus in s = vt, where s is the space passed over in time t 



ILLUSTRATIONS OF DIFFERENTIALS 15 

by an object moving with constant velocity v, s changes 
uniformly. 

In s = J gP, where the object moves with constant accel- 
eration g, s changes non-uniformly. 

9. Differentials. — The differentials of variables that 
change uniformly with respect to each other are their corre- 
sponding increments; that is, their actual changes. 

The differentials of variables that change non-uniformly 
with respect to each other are what would he their corre- 
sponding increments if, at the corresponding values con- 
sidered, the change of each became and continued uniform. 

As with increments, the differentials will be positive or 
negative according as the variables are increasing or de- 
creasing. 

The differential of a variable is denoted by putting the 
letter d before it; thus, dx, read '' differential x," is the 
symbol for the differential of x. The differential of a vari- 
able or function consisting of more than a single letter is 
indicated by the letter d before a parenthesis enclosing the 
variable or function; thus, d{x'^), dimx-^b), d{f{x)), 
denote the differentials of x'^, mx + h, and/(x), respectively. 

10. Illustrations of Differentials. — (a) Suppose a rec- 
tangle, with constant altitude, is changing by the base in- 
creasing. If when the base is 

AB its increase is BM, then 
d (base) = BM, and d (rectangle) 
= BMNC. 

Here the variables change uni- 
formly with respect to each other, 
hence their differentials are their 
corresponding increments. 

(6) Conceive a right triangle, with variable base and 
altitude, is changing by the altitude moving uniformly to 
the right. If when the base is AB its increment is BM, then 
the increment of the triangle will be BMDC. But if the 




16 



DIFFERENTIAL CALCULUS 




increase of the triangle became uniform at the value ABCy 
the increment of the triangle in the same time would evi- 
dently be BMNC; hence, BMNC and BM may be taken as 
the differentials of the triangle and 
of the base, where the base is AB. 

In this case the triangle changes 
non-uniformly with respect to its 
base, so its differential is what would 
he its increment if, at the value con- 
sidered, the change became uniform. 
Since the base changes uniformly, its 
differential is its actual increment. 
Here increment of triangle ABC = 
d (triangle ABC) -f triangle CND, 
while A (base) = d (base). If the 
change of a variable be uniform, any actual increment may 
be taken as its differential. If time be considered, the in- 
terval of time, though arbitrary, must be the same for a 
function as for its variable. 

(c) Let the curve OPn be the locus oi y = f{x), referred 
to the axes OX and OY. Conceive the area between OX 
and the curve as traced by 
the ordinate of the curve 
moving uniformly to the 
right. Let z denote this 
area, and let MMi be A.x 
reckoned from the value 
OM = x; then MMiPiP 
= ^z. But if the increase 
of z became uniform at 
the value OMP, its incre- 
ment in the same interval 
would be MMiDP; hence 
taken as 
X = OM, 




MMi and MMiDP may be 
the differentials of x and z respectively, when 



ILLUSTRATIONS OF DIFFERENTIALS 



17 



Hence dz = MMiDP = MPdx = ydx, 

which shows that area z is changing y times as fast as x. 
Here Az = dz + area PDPi. 

It is seen here that while the actual change in the area 
does not admit of an exact geometrical expression, the 
differential of the area, being a rectangle, is exactly and 
simply expressed. It will be shown further on how by 
Integration an exact expression for the area itself is obtained 
from this expression for the differential. 

Note. — Historically the Calculus originated through the 
efforts to obtain the exact area of figures bounded by curves, 
mathematics up to that time having furnished no method 
applicable to all curves whose equations were known. 

It is true too that historically the method of Integration 
was discovered before the method of Differentiation was 
developed. The Differential Calculus arose through the 
problem of determining the direction of the tangent at any 
point of a curve. (See Note, Art. 75.) 

(d) Let OPn be the locus oi y - f {x) and s the length 
from along the curve. Suppose the point (x, y) to move 
along the curve to P and thence 
along the tangent at that point. 
Then at the value x = OM, the 
change of x and y would become 
uniform with respect to each 
other, as the point (x, y) would 
be moving along a straight line. 
The change of s would become 
uniform also with respect to both 
X and y. As x is the independent 
variable it may be taken to vary 
uniformly, making PD or dx = 
Ax or MM I, the actual change in x as the point moves along 
the curve from P to Pi. Then dy is DT, the correspondmg 




18 DIFFERENTIAL CALCULUS 

uniform change of y, and ds is PT, the corresponding uniform 
change of s. It is evident that while dx = Ax, dy is not 
equal to Ai/ and ds is not equal to As. When, and only when, 
the locus is a straight line will dy = Ay and ds = As, after dx 
has been taken equal to Ax. 

It should be noted that it is not essential that dx should be 
made equal to Aa:, for dx may be taken as any value other 
than zero, and then dy will be the perpendicular distance 
from the end of dx to the tangent and ds will be the distance 
from the point (x, y) along the tangent to end of dy. From 
figure, {dsY = {dxY + {dyY. 

11. Rate, Slope, and Velocity. — The differential triangle 

PDT in figure for {d) Art. 10, gives -^^ = tan = slope of the 

dv 
curve y= f(x) at point {x, y) , and -^ is the ratio of the 

change of y to the change of x at the point {x, y), or for any 

dii 
corresponding values of x and y, and -r- is called the rate 

of y with respect to x. 

-r-^ is the average slope or the average rate of change of y 

with respect to x, while the point {x, y) moves over As on 
the curve or while x and y take successive values over any 
range. 

If s —f{t), where s denotes distance from some origin, 

and t, the time elapsed, then -7- is the rate of change of s 

with respect to t, what is called velocity, speed, or rate of 

ds 
motion: v = ^-' 
dt 

In the case of uniform motion in a straight or curved path, 

t? = 7 = -r- = -Tr=a constant. In the case of non-uniform 
t At dt 

ds 
or variable motion, v = -r- = a. variable. 
dt 



RATE, SPEED, AND ACCELERATION 19 

In figure for (d) Art. 10, it is seen that (dsy = {dxY + {dyf) 
dividing by (..)^ gJ = (|)V(S;; 

velocity of a point in its path is resultant velocity, 



'=S=V(f+(S)'=^s 



>y\ 



a:-component i^Vx = -7: = velocity parallel to x-axis, 

i/-component mvy = -£ = velocity parallel to ?/-axis; 
, , dy dy / dx dy dx ^ , ^ , 

dy dy / ds dy ds . 

dx dx /ds dx ds 

dv 
It appears that -p , the rate of y with respect to x, is the ratio 

of the time rate of y to the time rate of x. 

These expressions for velocity and their relations include 
the case in which the motion is uniform or variable along a 
straight line. 

12. Rate, Speed, and Acceleration. — Acceleration is 
rate of change of speed or velocity. Hence, if the speed is 

changing, -3- , the time rate of change of speed, is called the 

acceleration along the path, or the tangential acceleration, 
and will be denoted by at. The total acceleration a is equal 
to at, when the path is a straight line; otherwise, they are 
not equal. It is desirable to distinguish between speed and 
velocity. A body is in motion relative to some other body 
when its position is changing with respect to that other. 




20 DIFFERENTIAL CALCULUS 

Change of position involves change of distance or of direction 
or of both distance and direction. If a point moves con- 
tinuously in the same direction, the path is a straight line; 
if the direction is continuously changing, the path is a curved 
line. The direction of motion at any point of a curvilinear 
path is the direction of the tangent at that point, and from 
one point to another the direction of motion changes through 

the angle between the two tan- 
gents. Thus from Pi to P2 the 
direction changes through angle 
4>. When the position of a point 
changes the displacement takes 
place along some continuous 
path, straight or curved, and a 
certain time elapses. The rate at which the change of posi- 
tion takes place is the velocity of the point. 

If the point moves so that equal distances are passed over 
in equal intervals of time, the motion is uniform and the 
point has constant speed, whether the path is straight or 
curved. If the direction also is constant, that is, if the 
path is a straight line, the point has constant velocity. Thus 
there is uniform motion with constant speed either in a 
straight line or in a curved Hne, but there is uniform or 
constant velocity in a straight line only. 

The extremity of either hand of a clock moves in a circular 
path over equal distances in equal intervals of time, but its 
direction is continuously changing. The motion is uniform 
and the speed constant, but the velocity is not constant since 
the direction is variable. Hence, a body may move in a 
circle with constant speed and yet its velocity is variable. 
In this case the acceleration at along the tangent is zero, 
while the total acceleration a, the rate of change of the 
velocity, is normal, directed towards the center, and has a 
constant value depending upon the speed and radius. (This 
value will be derived later.) 



ILLUSTRATIONS 21 

The term speed thus denotes the magnitude of a velocity. 
However, the term velocity itself is ordinarily used in the 
sense of speed as weU as in the strict sense of speed and 
direction. In the great majority of cases the direction is 
assumed to be known, and the magnitude of the velocity is 
what is in question. 

Note. — A velocity having both magnitude and direction 
is what is called a vector quantity and can be represented by 
a straight hne having the direction of the velocity and a 
length denoting its magnitude. Hence the sides of the tri- 
angle PTD in figure for (d) Art. 10, may be taken to repre- 
sent the resultant velocity v and its components Vx and Vy. 

13. Rate and Flexion. — Flexion has been adopted by 

some writers as a term for the rate of change of slope. Hence, 

when the slope changes, and it always does except for a 

dwi 
straight line, -p , the rate of change of the slope with respect 

to X will be called the flexion of the curve and will be denoted 

by h, from the word bend. When the velocity and the slope 

are uniform, there is no acceleration and no flexion; that is, 

dv ^ , dm ^ 
-rr = and ^- = 0. 
dt dx 

14. Illustrations. — Consider the established equations 

of motion: 

^• 
s = vt ov V = -} 

L 

V = gt = S2t(at = g = 32 ft. per sec. per sec. approximately); 
s = ^gt^ = IQ t\ 

When the motion is uniform the velocity or speed is the 
whole distance divided by the whole time ; or any increment 
of the distance divided by the corresponding increment of 
the time is the velocity at any point, and it is the same as 
at any other point, since it is constant. 

s ^s ds , , 

/. 2^ = 7 = TT=Tr = a constant. 
t Ai dt 



22 DIFFERENTIAL CALCULUS 

In the case of variable motion the whole distance divided 

by the whole time gives the average velocity over the whole 

distance; or any increment of the distance divided by the 

corresponding increment of the time gives the average 

velocity over that increment of the distance. The velocity 

at any point is now given by the distance that would be gone 

over in any time divided by that time, if at the point the 

motion became and continued uniform or the velocity 

became constant. 

ds 
Thus ^ = -3-. = 32 ^ gives v = 32 ft. per sec. at the end 

of the first second ; and means that the distance in the next 
second would be 32 ft., if at the end of the first second the 
velocity became constant. 

As a matter of fact, s = 16 ^^ gives 16 feet for the distance 
in the first second, and 48 feet for the distance actually 
passed over in the next second. This variation of distance 
is of course due to the velocity being constantly accelerated. 

So when it is said that a train at any point is moving at 
sixty miles per hour, it is not asserted that it will actually go 
sixty miles in the next hour; but what is imphed is, that the 
train would go sixty miles in any hour if from that point it 
continued to move with unchanged velocity. 

Therefore, in ordinary language, variable velocity is ex- 
pressed by the differential of the distance divided by the 

ds 
differential of the time; that is, by -77- 

In the case above, 

ds _ 60 miles _ 1 mile _ 88 feet . 
~di 1 hour 1 min. 1 sec. 

thus dt may be taken as any value other than zero, if the 
corresponding value of ds is taken. 



EXERCISE II 23 



EXERCISE n. 

1. u = 2 X. Show graphically the change in u when x is given an 
increment, by taking x as the base of a variable rectangle of altitude 2, 
and iX as the area. Is the change uniform for m ? 

2. u = x^. (a) Show graphically the change in u when x is given an 
increment, by taking x as the side of a variable square, and u as the 
area. Show graphically the change in it if the change became uniform. 
(6) Show same when x is taken as the base of a variable right triangle of 
altitude 2 x, and u as the area. Show the change in w if the change 
became uniform. 

3. V = x^. Show graphically the change in V when x is given an 
increment, if the change in V became uniform; x being the side of a 
variable cube, and V the volume of the cube. 

4. If a body is moving with uniform velocity and passes over 1000 
feet in 10 seconds, what is its velocity at any point? If distance is 
taken as axis of ordinates and time as axis of abscissas, what would the 
slope of the graph be ? 

5. s = 16 i^. Compute the values of s when t = 1,2, 1.1, 1.01, 1.001. 
Get the average velocity between t = 1 and t = 2, between t = 1 

and t = 1.1, between t = 1 and t = 1.01, between t = 1 and t = 1.001. 
From V = 32 t, get the velocity at f = 1 and compare average 
velocities with it. What would be the distance passed over in the 
second second, if at the end of the first the velocity became uniform? 
What is the actual distance passed over in the second second ? Which 
is the increment ? Which is the differential ? 

6. If a ship is sailing northeast at 10 knots, what is its northerly 
rate of motion ? What is its easterly rate ? 

If it is sailing S. 30° W. at 10 knots, what is its southerly rate ? What 
is its westerly rate ? 

7. If the grade of a road is such that the rise is 52.8 feet in every 
mile, what is the slope? 

8. If the grade of a road is continuously changing, the average slope 
is given by what ? The slope at any point would be the slope of what ? 



CHAPTER II. 

DIFFERENTIATION. DERIVATIVES. LIMITS. 

15. Derivative. — The ratio of the differential of a 

function of a single variable to that of the variable is called 

dv 
the derivative of the function. Thus ^ denotes the deriva- 

dx 

tive of ?/ as a function of x. Since the derivative may vary 

with X, as the slope of a curve varies from point to point, it 

is, in general, itself a function of x] hence, the derivative of 

f{x) is appropriately denoted hy f{x), and is often called 

the derived function. So 

K 2/=/(x), 

.-. dy = f(x)dx. 

Since dy = f (x) dx, the derivative is also called the differ- 

du 
ential coefficient. The derivative -r- is sometimes denoted 

by D^y. 

In the case of a curve the derivative is the slope, in the case 
of motion it is the velocity, speed, or rate of motion; hi every 
case it is the rate of change of the function with resj^ect to the 
argument or variable. 

Examples. — 

li y = fix) = mx + b, the derivative -f- = f (x) = m, 

24 



LIMITS 25 

ds 
If s = /(O = vt, the derivative ^ = f(t) = v, 

dv 
li V = f{{) = gt, the derivative -^ = f{t) = g. 

Here m, v, and g are constants. 

16. Differentiation. — The operation of finding the 
differential of the function in terms of the differential of the 
argument, or the equivalent operation of finding the deriva- 
tive, is called differentiation. The sign of differentiation is 
the letter d; thus d in the expression d (x^) indicates the 

operation of finding the differential of x^, and in -y- {x^) , that of 

finding the derivative. D^y, j- , and /' (x) each denote the 

derivative of ?/ as a function of x. 

The general method of getting the derivative oi y = f (x) 

is by finding the limit of the ratio of the increments of y and 

X as they are diminished towards zero as a hmit; for the 

limit which the ratio approaches, when defined to be the 

dv 

derivative, can be shown to be 3— 

' dx 

17. Limits. — The student has been made acquainted 
in Geometry with the notion and use of limits; for exam- 
ple, the area or the circumference of the circle, as the 
limit of the area or the perimeter of the inscribed and cir- 
cumscribed polygons, when the number of sides increases 
without limit, or when the length of the side approaches zero 
as a limit. A precise statement of a limit as used in the 
Calculus is as follows : 

When the difference between a variable and a constant becomes 
and remains less, in absolute value, than any assigned positive 
quantity, however small, then the constant is the limit of the 
variable. 

If X is the variable and a is the limit, the notation is 
lim X = a, or X = a, or lim {a — x) = 0, or (a — x) = 0; 



26 DIFFERENTIAL CALCULUS 

in which = is the symbol for approaches as a limit. When 
the hmit of a variable is zero, the variable is an infinitesimal. 
The difference between any variable and its limit is always 
an infinitesimal. 

18. Theorems of Limits. — The elementary theorems of 
hmits are: 

1. If two variables are equal, their limits are equal. 

2. The hmit of the sum or product of a constant and a 
variable is the sum or product of the constant and the limit 
of the variable. 

3. The limit of the variable sum or product of two or 
more variables is the sum or product of their limits. 

4. The limit of the variable quotient of two variables is 
the quotient of their limits, except when the limit of the divisor 
is zero. (See Note, Art. 20, for proof.) 

Note. — The Differential Calculus solves such limits as 
the exceptional case just stated. 

19. Derivative as a Limit. — The limit of a variable, as 
z, is often written It (z). 



Limr^l, or It ^, denotes It(^) 



when Ax = 0. 



dv A'Z/ 

In defining -p as a rate (Art. 11), it is stated that -r^ is the 

average slope, or average rate of change of y with respect to 
x, over the range Ax. As has been given (Art. 7), 

Ay ^ Af(x) ^ fix + Ax)-^f{x) ^ 

Ax Ax Ax 

where y = f (x) and x is any value of x. It remains to be 
shown that 

lim \^] = i^f±±MjzI(d = ± =/'(^). 

ax=oLAa;J az=o Ax dx 

(a) By rates without the aid of a locus. Let time rates be 
used and let t, x, and y denote any corresponding values of 



REMARKS. FUNCTION OF A FUNCTION 27 

t,'x, and y, from which A^, A;r, and A?/ are reckoned. Since 

Aiy • Ai/ 

-rj is the average rate of y over interval At/, -ry is the time 

rate of 2/ at a value of y between y and 2/ + A?/; 

A^ _ ( time rate of y at the value oi y ( . . 
At I from which A^ is reckoned. ) 

^, , Aa; _ ( time rate of a: at the value of x ) .^. 

A^ I from which Ax is reckoned. ) 



Dividing (1) by (2), there results, 

Ax=o L^^J the time-rate of a; dt / dt dx 

(Compare Art. 11.' 



,. 1 — i^ I _ the time-rate of ^ _dy jdx _dy 



Thus in showing the derivative as a limit, it appears that the 
derivative of a function expresses the ratio of the rate of 
change of the function to that of its variable. It is evident 
that a function is an increasing or a decreasing function 
according as its derivative is positive or negative. 

In the above derivation in place of time-rates, the rate of 
any other variable of which x and y are functions could be 
used. 

Remarks. Function of a Function. — It should be 
noted that x and y being taken as functions of a third 
variable ^, to every value of this auxiliary variable there 
corresponds a yair of values of x and ?/, so y is indirectly 
determined as a function of x. The derivative of 2/ as a 
function of x mediately through t is, as shown; 

^ — ^ /do; 
dx dt I dt 

c, 1 . . dy dy dx 

Solving gives di^di'Tt' 

and this gives the formula for the derivative of the func- 
tion of a function. For if y is directly given as a 
function of x, and x as a function of t, then y is said to 



28 DIFFERENTIAL CALCULUS 

be a function of a function of t, as it is given as -a 
function of t mediately through x. If 2/ is a given function 
of z, and z a given function of x, then 2/ is a function of a 
function of x, and the formula for the derivative of y is 

dy_ _ dy dz 
dx dz dx 

Functions of functions often occur and there may be 
several intermediate variables such as z in above case. A 
function, as/ (x), is defined to be continuous for the value a 
of X, or, more simply, continuous at a, if / (a) is a definite 
finite number, and if lim f{x) = f (a) ; that is, if hm / (x) = 

f (lim x) . By this definition the elementary functions of a 
single variable are continuous for all values of the variable 
except those for which a function becomes infinite. 

A concrete case in everyday experience of a function of a 
function is the change in length of a metal bar as the tem- 
perature changes with time. Here the length is a function 
of the temperature, and the temperature is a function of the 
time; hence the length is a function of a function of the time. 
The length, being directly a function of the temperature, is 
indirectly a function of the time through the temperature, 
which is directly a function of the time. 

The rate of change of length per second is equal to the 
product of the rate of change of length per degree and the 
rate of change of temperature per second. If I, T, and t 
denote the length, temperature, and time, respectively, then, 
in accordance with the formula, 

dl^dl_ dT 
dt dT' dt' 

If the length and temperature are taken as changing each 
directly with the time, then the rate of change of the length 
per degree is equal to the rate of change of length per second 
divided by the rate of change of temperature per second. 



curve. 



REMARKS. FUNCTION OF A FUNCTION 29 

The formula would be 

dl^^dl /dT 
dT " dt/ dt ' 
which may be obtained from the other formula by solving; 
or the first may be obtained from this. As all variables 
change with time, that is, are functions of time, time rates 
are most common. 

(6) To show geometrically 

lini r^l = #^ = slope of 

Let OPn be the locus oi y = f{x), PP^S a secant, and PT 
a tangent at P. If arc OP = s, arc PPi = As. Let 

OM = X, MMi = t^x, then MP = y, DPi = ^y. 
Hence 

— ^ = 7^ = slope of secant PPiS. 
Ax PD 

Conceive the secant PPiS to be re- 
volved about P so that arc PP\ 
(= As) = 0; then ^x = 0, A?/ = 0, and 
the slope of the secant = the slope 
of the tangent at P. 

Hence hm hr^ = :/ = slope of the curve y = f{x) at 
Ax=o L^^J (ix 

point {x, y). 

dv 
The limit is thus shown to be the derivative, whether -j^ 

dx 

is regarded as only a symbol for the limit of the ratio of the 

increments of y and x, or as that limit and also a definite 

ratio itself of the differentials of y and x. 

Corollary. — If when Aa; = 0, -r^ varies, the locus of 

Ay 
y = f {x) is a curved line; otherwise, if —- is constant, the 




30 DIFFERENTIAL CALCULUS 

locus is a straight line coincident with the tangent, and 

^ = -1- . So for a straight line, the ratio of the increments 

of y and x, being constant, does not approach a limit as Ax 
approaches zero, and the derivative is the constant ratio 

^x dx 

In general, ^ will approach a finite limit except where the 

locus is perpendicular or parallel to the :r-axis, when the 
slope is infinite or zero. On special curves where there are 
two tangents at a point, the limit is not definite. (See Note, 
Art. 80.) 

Note. — This definition of the derivative as the limit of 
the ratio of the increments of y and x as • they converge 
towards zero is the fundamental conception of the Differential 
Calculus by the method of limits. 

In this method since A?/ and Ax are variables each approach- 
ing zero as a limit, they are infinitesimals, for any variable 
with zero as a limit is defined to be an infinitesimal. If dx 
is taken as always equal to Ax, then, except when the locus 
is a straight line, dy will always differ from Ay; and dx, dy. 
Ay and (A^ — dy) are infinitesimals when Ax approaches 
zero, for they each approach zero as Ax approaches zero. 
However, dx may be taken as any increment of x and, when 
X is the independent variable, may be made a finite constant, 
for X may be considered as changing uniformly by finite 
increments; but then, except for a straight line, dy is variable 
though finite. When Ax is infinitesimal any particular value 
of Ax may be taken as constant, for any particular val ue of 
an infinitesimal is a fixed finite quantity, small or large as the 
case may be. Whether dy and dx are infinitesimals or finite 
quantities, their ratio for any particular value of the variable 
is, in general, constant; and it is their ratio that is important. 

Both ways of regarding differentials are useful. Finite 



ILLUSTRATIVE EXAMPLES 31 

differentials are desirable for their simplicity, especially to 
make the differential of the independent variable constant. 
But when Integration is regarded as finding the limit of a 
sum, as will be shown later, differentials are necessarily 
infinitesimal. 

One advantage in making dx infinitesimal and taking it 
very small is that A^ is then very nearly equal to dy, and so 
instead of computing A?/ in some investigation the simpler 
and easily found dy may be taken for it. In practical work 
dx and dy are usually taken very small quantities, but it is 
their ratio that is of importance. In this connection it should 
be borne in mind that, however small a quantity may be, 
it is not an infinitesimal as defined in the Calculus, unless 
it is a variable approaching zero as a limit. 

20. Illustrative Examples. — In these examples, as 
elsewhere, the letter symbol for the argument in general is 
used as the symbol also for some particular value of the 
argument; this double use of the symbol making for con- 
ciseness and generahty. 

Example 1. — Let the function to be differentiated be 

y = x\ _^ (1) 

2/ + A?/ = (a: + ^xf = x^^2x t^x^ b,x , (2) 

A?/ = 2 x^x + Ax^, (3) 

g = 2x + Ax, (4) 

^ = ^ = lim r^l = lim (2x + Ax) = 2x; (5) 

.*. dy = 2x dx. (6) 

The actual change of y corresponding to any change of x is 
givai by (3). The average rate of change of y from any 
value of X to X 4" Ax, or the average slope of the curve over 
that range, is given by (4). The rate of change of y with 
respect to x at any value of x, or the slbpe of the curve at 



32 DIFFERENTIAL CALCULUS 

any point {x, y), is given by (5). What would be the change 
of y for any change of x, if at any value of x the change of y 
became uniform, is given by (6) ; and it shows that, at any 
point {x, y), 2/ is changing 2 x times as fast as x is changing. 
Example 2. — Let the function to be differentiated be 

s= 16^2^ (1) 

s + As= 16(^ + A02= l^{t^ + 2t^t + ~^i), (2) 

As = 32 ^ A^ + 16 A^^, (3) 

(4) 

(5) 
(6) 

(5) 
(7) 
(8) 

(9) 
(10) 

The distance s passed over by a body falhng from rest in any 
time t is given by (1). The actual distance passed over in 
time A^, after any time t, is given by (3). The average time- 
rate of the distance, or the average velocity from s to s + As, 
is given by (4). The time-rate of the distance, or the 
velocity at end of any time t, is given by (5). What would 
be the distance passed over in time d^ (= A^, if at end of 
time t the body moved on with unchanged velocity, is given 
by (6). The actual change of the velocity in time Ai is given 
by (8). That the velocity is changing uniformly is shown 
by (9), since the ratio of the two increments is constant. 



As 

A^ 


= 32 ^ + 16 A^, 


V 


ds 
dt 


= limr^'] = 32^; 


.-. ds 


= 32tdt. 


the function to be differentiated be 




V -- 


= 32^. 


v-\- H^v ■■ 


= 32 (^ + AO, 




. Av. 


= 32 A^, 




Av 
At ' 


= 32, 




a 


-""'-dt-Ii-^^' 



ILLUSTRATIVE EXAMPLES 33 

The rate of change of the velocity or speed, the acceleration 

a or at, is given and shown to be constant by (10). 

Av 
It is to be noted that ^, being a constant ratio by (9), 

does not approach a limit; hence, the derivative -^ is equal 

to — and is therefore constant acceleration. (See Corollary y 

(h), Art. 19.) 
Note. — If s = 16 ^2 be represented graphically by a curve, 

■ds 
then the slope of the curve is m = -r. = S2 t = v, the veloc- 
ity; and the flexion of the curve is 6 = -rr = -7: = 32 = a«, 

the acceleration. 

Example 3. — Let the function to be differentiated be: 

y = mx + h. (1) 

y -\- Ay = m {x -\- Ax) + 6 = mx -\-m* Ax -{-h, (2) 

Ay = m- Ax, (3) 

!=£-. (^) 

.*. dy = mdx. (6) 

Here again the ratio of the increments, being shown by (4) 

to be a constant m, does not approach a hmit; hence, as 

dv All 
shown by (5) the derivative j~ = T~ = ^j ^tie constant 

slope of the line y = mx + h. 

That the ordinate is changing m times as fast as the 
abscissa is shown by (6). 

It is evident that for a linear function not only is the ratio 
of the increments the derivative, but the increments are the 
differentials as defined. 



34 



DIFFERENTIAL CALCULUS 



Example 4. — Let the function to be differentiated be 

1 , 

y = - or xy = 1. 

X 



(1) 



(x + Ax) (y-{-Ay) = xy-{-x\y-\-yAx + Ax Ay = 1, . (2) 
xAy + yAx-\-AxAy=0, or (x + Ax) Ay = —y Axj (3) 

^y y 







, average slope over Ax, (4) 

y 



= tan (j), 



Ax X + Ax 

Ax=o V x-\- AxJ 
slope at any point (x, y) ; (5) 

/. dy=-ldx, (6) 

showing that the . oi y is - times the 



increase 



increase 



of X, at any point {x, y). 



decrease 

Example 5. — In compressing air, if the 
temperature of the air is kept con- 
stant, the pressure and the volume 
are connected by the relation pV = 
constant. To find the rate of change 
of the pressure with respect to the 

dij 
volume, that is, the derivative 7^- 

dV 

Let pV = K, (1) o"vt 

(p + Ap) (V + AV) = pV+pAV + VAp + ApAV = K, (2) 

pAV-{-VAp + ApAV = 0, or (V + AV) Ap = -pAF, (3) 

Ap _ p 

AV~~V+AV' 
average rate of change from F to F + A T , ^ 

^ =]!??„ [If] =ii'?o (- F+af) = - f 

rate of change for any corresponding values of p and Vi 



(4) 



(5) 



ILLUSTRATIVE EXAMPLES 35 

.*. dv = — —dV, showing that the . of pressure is 

^ V increase 

p ,. ,, increase r- i ^ ^^ ^ 

^ times the , oi volume at any corresponding values 

of pressure and volume. 

Example 6. — Let M be the mass of a body, V its volume, 
and p its density; then, 

AM ^M ^ 
AV V ^' 
the density at any point, when the body is of uniform 
density; 

,. AM dM 
AV=oAV dV ^' 
the density at any point, when the density varies from point 
to point. Here when the body is not homogeneous, the 

density being variable, -r-y. is the average density of the 

portion of mass, AM; while the derivative, -tj^, expresses 

the density at a point of the body whether the density is 
variable or uniform. 

Note, — In regard to lim hv = :i^, the derivative of y 

as a function of x, it is important to note that, since the 
Hmit of the divisor is zero, it is wrong to write 

j.^ rA,1 InnAj/ ^ 

Aa;=o L^^J hm Ax 
This case is specially excepted in Theorem 4, Art. 18. To 
prove the Theorem 4, Art. 18: 

Since y = -'X, 

X 

lim y = Hm - • lim x, by Theorem 3, 

X 

,. y limy ..-,. 
\ um - = r: — - , if lim X IS not zero, 
X max 



36 DIFFERENTIAL CALCULUS 

When lim x is zero, division by it is inadmissible by the laws 
of Algebra. If lim x were zero and lim y not zero, then - 

X 

is infinite and has no hmit; hence the exception in Theorem 

y 
4. The notation lim - = oo , if so written, means that, as x 

x=OX ' 

approaches zero as a Umit, - increases without hmit; that is, 

X 

the limit is non-existent. 

Infinity or an infinite quantity is not a limit, and the 
symbol oo means a variable increasing without limit. 

In Example 4, where y = -, - = — . Here where lim x 

^ x' X x^ 

y . 
is zero and lim y is not zero, - is infinite, having no limit. 

X 

In Example 5, the limit — ^ is finite for finite values of p 
and V. From V = y and ^ = r—, p = oo asF = 0; hence 
as lim V is zero and lim p is not zero, ^ is infinite; 



Ay " ~ 7-f AF ^^ i^fi^ite as F = 0, 



and lim -r^ is non-existent when V is infinitesimal. 

Av 
In Example 3, where y = mx + h, Ay = m Ax, and -i^ = m. 

If Ax = 0, At/ = 0, but their ratio is constant and ap- 
proaches no limit. Since Ay = mAx, the law of change of 
the variables is known and the ratio of two infinitesimals is 
a finite constant. 

In Example 1, where y = x^, 

Here the limit of the ratio of two infinitesimals is a finite 
constant for any particular finite value of x; but, as x may 



Ay 

Ay = 2xAx + Ax^, -r-^ = 2 x + Ax, .*. lim 

Zax Ax=0 



LIMIT OF INFINITESIMAL ARC AND CHORD 37 

have any value, the hmit of the ratio may be zero, finite, or 
non-existent. 

Thus it is seen that, no matter how small two quantities 
may be, their ratio may be either small or large; and that, if 
the two quantities are variables both with zero as their limit, 
the limit of their ratio may be either finite, zero, or non- 
existent, but is not 0/0. (See Art. 219.) 

To find hm -j-^ , as in the illustrative examples, the limit 

of an equal variable is found; which limit is, in general, 
determinate and not identical with the indeterminate expres- 
sion 0/0. In certain cases the limit of the ratio of two 
infinitesimals is found by finding the limit of some other 
variable which, though not equal to the ratio, has the same 
limit. Examples of such cases will be given further on. 
(See Art. 70, Art. 77.) 

21. Replacement Theorem. — The limit of the ratio of 
two variables is the same when either variable is replaced by any 
other variable the limit of whose ratio to the one replaced is unity. 

Let 0, 01, (f), and 0], be any four variables, so that 

c. (1) 



It 


6 
01 


1, 


It 


01 


1, and 


< 







e 




'01 


<t>i_ 
*<Ai' 


_ ^1 
01* 



01 ' 


<t>i. 




< 




= it 


01 
01* 


< 


■«? 


= lt 





by(l) 

in which is replaced by ^i, and by 0i, but the limit of the 
two variables is the same. 

22. Limit of Infinitesimal Arc and Chord. — The limit of 
the ratio of an infinitesimal arc of any plane curve to its chord 
is unity. 

Since s (Art. 19 (6), figure) is a function of x, 

••• «^ = ?- (1) 

^x dx ^ ^ 



38 DIFFERENTIAL CALCULUS 

But It ^^Q^^ ^^1 = It sec DPPi = sec DPT = ^ • (2) 

Dividing (1) by (2), l^^[^jj2p^] = l. (3) 

It follows from Art. 21 that in a limit an infinitesimal arc 
may be replaced by its chord. 

ALGEBRAIC FUNCTIONS. 

23. Formulas and Rules for Differentiation. — By the 

general method any function can be differentiated, but it is 

usually more directly done by formulas or rules estabhshed 

by the general method or by other methods. 

In the following formulas u, v, y, and z denote variable 

quantities, functions of x; and a, c, and n, constant quan- 

'^ c? " 
tities. If in the formulas -r- or '^ Da; " be substituted for 

^'^," and in the rules '^derivative" be substituted for differ- 
ential, they are still valid. 

[I] If y = X, dy = dx. 

The differentials of equals are equal. 

[II] d(a)^0. 
The differential of a constant is zero. 

[III] d (v + y + • . . - ;5 + c) = di; + dy + . • - dz. 
The differential of a polynomial is the sum of the differentials 

of its terms. 

[IV] d {ax) = a dx. 

The differential of the product of a constant and a variable is 
the product of the constant and the differential of the variable. 
[Ya] d (uy) = ydu + udy. 

The differential of the product of two variables is the sum of 
the products of each variable by the differential of the other, 
[Vft] d (uyz , , , ) = (yz . . , )du -\- (uz , , , )dy 

-\r {uy , , , )dz+ ' ' * ' 



DERIVATION OF [I] 



39 



The differential of the product of any number of variables 

is the sum of the products of the differential of each by all the 

vest 

.^^', ,(N\ DdN-NdD 
[VI] dl^^j = ^ 

The differential of a fraction is the denominator by the 
differential of the numerator minus the numerator by the 
differential of the denominator, divided by the square of the 
denominator. 



nx 



dx. 



[VII] d(x^) 

The differential of a variable with a constant exponent is the 
product of the exponent and the variable with the exponent less 
one by the differential of the variable. 

24. Derivation of [I]. — If ?/ is continuously equal to x, 
it is evident that y and x must change at equal rates; 
that is, 

dy _ dx 
dx dx' 
dx 



dy = dx. 



Since 



dx 



1, the rate of x is the unit 



rate, so in general the rate of a variable 
with respect to itself is unity, or the 
derivative of / (x), when / (x) is x, is 
one. 

Geometrically the locus of y = x is the straight line 

through origin making angle 4> = j with a;-axis. 




jy-dy 



tan (^ 



!^x 



1, 



1 • A-?/ . ^ ^ ^y dy ^ 

and smce -r^ is constant, -r^ = ~ = \. 
Ilx I\x ax 



dy = dx. 



For examples of [I], if y"^ = 2px, d(y^) = d{2px); or if 

X2 + ^2 = (j2^ ^ (^2 + ^2) == ^ (^2) = 0. 



40 DIFFERENTIAL CALCULUS 

25. Derivation of [II]. — By definition the value of a 
constant is fixed, therefore the rate of a constant is zero; 
that is, 

^ = 0, .-. da = 0. 
dx 

li y = a, a change in x makes no change in y, hence 

Ay . Ay . 

Ay = 0, /. -r^ = 0, and since —■ is constant, 

Ax dx * ^ 

Geometrically the slope of 2/ = ot (a Hne parallel to x-axis) 
is at every point zero. 

26. Derivation of [III]. — It is manifest that the rate of 
the sum of v •\- y -\- • • • — 2! + cis equal to the sum of 
the rates of its parts, v, y, . . . —z and c; that is, 

d{v -j-y + ' ' ' — z + c) _dv _L_dy ^ ^ _^i^ 
dx dx dx dx dx 

Multiplying by dx, since dc = 0, the result is [III]. The 
rule shows that differentials are summed like any other 
algebraic quantities. For example, 

d (6V ± aV - a'^') = d {¥x'') ± d (aY) - d (a^h^). . 

27. Derivation of [IV]. — Since A (ax) = aAx, the ratio 
of the increments is constant and ax changes uniformly 

with respect to x. Hence by definition 

^ — ^ of differentials d (ax) = adx. li y = ax, 

dv 

-^ = a, slope of line. Geometrically, if 



dz 



X MdxM, 

z = axhe area of a rectangle of base x and 
altitude a, then the rectangle MPPiMi is the change of z 
made by a change Ax(= dx) of x, and being a uniform change 
is the differential of z, 

.*. dz = d (ax) = a dx. 



DERIVATION OF [Vb] 



41 



^y 


XI 




1 

1 






p 






z=ui/ 


y 


y 

du 



For examples: 

d(2px)=2pdx, and d(-) = dl-x]= — 

[Va] will be seen to include [IV] as a special case. 

28. Derivation of [V^]. — Let z = uy; then z, a function 
of 1^ is a function of y also. Geometrically, let u and y be 
the base and altitude of a variable rectangle conceived as 
generated by the side y moving 

to the right and the upper base ^ 

u moving upward; then z = uy ^ 

is the area. If at the value 

OMPN, du = MMi, and dy = 

NNi, the differential of the area ^ ^ ^ 

is MMiDP + NPBNi, as that 

sum would be the change of the area of the rectangle due to 

the change of u and y, if at the value OMPN the change of 

its area became uniform. Hence dz = d {uy) = y du-\-udy. 

Here A^ = A {uy) = d {uy) + P DPiB, since that sum is the 

actual change of the area due to the change of u and y. 

It is to be noted that iiy = u, then the rectangle is a square 
and area z = u^, 

.*. d {u^) = udu-\-udu = 2udu. 

li y = a, z = au, dz = adu -\- uda = a du, since da = 0. 
Hence [IV] is a special case of [Va]. 

29. Derivation of [V^]. — To prove d {uyz) — yz du -\- 
uz dy + uy dz. If in [Vo], yz is put for y; 

d {uyz) = yzdu + ud {yz) 

= yz du -{• u {z dy -{- y dz) 
= yzdu + uz dy + uy dz. 

By repeating this process the rule is proved for any number 
of variables, li y = z = u, 

then d {uyz) = d {u^) = u^du-\- u^ du+'u'^ du = Su^du. 



To derive d{uyz) geometrically, let V = xyz = uyz. 



42 



DIFFERENTIAL CALCULUS 



z 






^■ 



■ 0, 




If a;, ?/, and z be the edges of a variable right parallelepiped 

conceived as generated by the face yz moving to the right, 

the face xz moving to the front, and the face xy moving 

2 upward, then the volume is 

the product of the three 
edges; that is, V = xyz. 

If at the value OP, dx = 
AAi, dy =BBi, and dz = Cd, 
the differential of the volume 
is PAi + PBi + PCi; as that 
sum would be the change of 
the volume of the parallel- 
opiped due to the change of 
X, y, and z, if at the value OP 
the change of its volume became uniform. Hence 

dV = d (xyz) — yzdx -\- xz dy + xy dz. 
Here 

AV = dV+ PNi + PLi + PMi + PPi, 

since that sum is the actual change of the volume due to the 
change of x, y, and z. li y = z = x, then the parallelopiped 
is a cube and V = x^, 

.'. d (x^) = x'^dx -\- x'^ dx -]r x^dx = '^ x^ dx. 

30. Derivation of [VI]. — Let z = ~ {x and y indepen- 

X 

dent), then zx = y. 

.'. d (zx) = xdz -^ zdx = dy, 
zdx 



by[Vc 



Solving^ 



dz = ^ 



diy]=^ 



X 

x[ _ xdy — ydx 

X " x^ 



Corollary. 



d(2U-^; 



DERIVATION OF [VII] 43 

. /a\ xda — adx adx . -. 

for d[-]= ^ = 2~» ^1^^® ^^ = ^- 

Va/ a 

„ ^ (x\ adx — xda dx . , „ 

for a - ; = — ' since aa = 0: 

\al 0? a 

hence, for a fraction with constant denominator, use 
[IV]. For another derivation of [VI], see Corollary of next 
Art. 31. 

31. Derivation of [VII]. — I. When the exponent is a 
positive integer. 

(a) If n is a positive integer, x"" = x • x - x ' to n factors; 
hence, 

d {x"^) = d{x'X - X to n factors) 

= a;"~i dx + x'^-i dx + a:"~^ dx -{- • • • 

to n terms, by [VJ, 
= nx""'^ dx. 



(h) By the general method. Let y = x"^. 

y -\- t^y = {x -{- l^xY= x''-\- nx''-'^ Ax + (terms 

with common factor Ao: ), 

Ly = nx"^"^ ^x + (terms with factor Lx ) 

Ai/ 

— ^ = nx""-^ + (terms with factor Ax), 

-^ = hm hr^ = nx"~\ .*. dy = nx"~i c?x. 
n. When the exponent is a positive fraction. 



by 

• Binomial 

Theorem. 



Let 


y = x'^, 


then 


yn= ^m^ 




.-. d(y-) = d(x^), 




nt/^"^ dy = mx"^'^ dx, 



, m x*"~^ , mx'^'^y 
n y'^~^ n y" 



44 DIFFERENTIAL CALCULUS 



\x-) = 



ax = — x"" otx* 

n x'^ n 



III. When the exponent is negative. 

Let y = x~", n being integral or fractional; then 2/ = — > 

•*• ^^ = ^ (i^) = " ?? ^""^ ^^ f^^'' ^^'•' '^'^- ^^' 
/. dy = d (x""*) = — nx~"~i dx. 

Corollary. — d [ - j = d {xy~'^) = i/"^ c^o; — xy~^ dy 

_ dx _xdy _ ydx — xdy .^^, 

~ y y^ ~ y^ 

Note. — A general proof of [VII] by logarithms, given 
further on (Art. 37), includes the case where the exponent is 
incommensurable. So the Formula or Rule is valid for any 
constant exponent. It is called the Power Formula and is 
of most frequent application. 

Examples. — ■ 

d (Vx) = d (x^) = - x~2 dx = — y=' 

^ 2Vx 

d(—7-=] = d {x~^) = — ^x~^^dx = 7=' 

WxJ 2 2Vx' 

d{x^^)= V2x'^^-'^dx,{= l.^Ux-^^^dx, approximately). 

d{x'^) = TTX'^-^ dx (= 3.1416 a;2-^^^^dx, approximately). 

d {(ax + 6) ") = n (ax + h) "-^ d (ax -{-h) =na (ax + b) ""^ dx, 

/. 4- ((«^ + ^) ") = na (ax + b) ^'K 
ax 

Note. — The last example may be seen to be an application 
of the formula for the derivative of a function of a function. 
For let y = (ax + 6)" and put z = ax + b, then y = z^; 
now 2/ is a function of z, and ;s is a function of x; that is, y is 



EXERCISE III 45 

a function of a function of x. The formula given in Remarks, 

Art. 19, is ^ = ^-^; 
' ax dz dx^ 

dy d ,. , ,. . d{z'') d(ax-\-h) ^ . 

dx dx^^ dz dx 

In applying Rule [VII], if all within the parenthesis, as 
{ax + 6), is regarded as the variable, the actual substitution 
of z may be dispensed with in getting the derivative of such 
functions. 

EXERCISE m. 

By one or more of the formulas I-VII differentiate: 

dy =d{Q x^) + d (4 x-^) - d (2 x-') + d {S x'^). 

dy^_^ ?_il_l?. 

dx ~ ^x V^ ^^ ^ ' 

2. 2/ = 3 x3 - 4 a:2 - 2. 

dy =d(3a;3) -d(4a:2) -d{2). 
dy = 9x^dx - Sxdx - 0; 

^ = 9x2 -8x = (9^ -8)x; 

that is, y changes (9 x — 8) a; times as fast as x. 

When x = —1, y is increasing at the rate of 17 to 1 of aj; 

X = ^, y is neither increasing nor decreasing; 

X = 0, y is neither increasing nor decreasing; 

X = I, y is decreasing at the rate of | to 1 of a;; 

X = I, y is changing at the same rate as x; 

X = —l,yis changing at the same rate as x] 

X = 2,yis increasing at the rate of 20 to 1 of x. 
Note. — In this way the meaning of each differential equation may 
be shown. 

3. t/ = (l+2a;2) (H-4a;3). dy = ix {1 +Sx + lOx^) dx. 

dy = (1 -h 2x2) d (1 + 4a;3) _^ (1 ^^x^)d (1 + 2a;2); 
or d^/ =£^(1 +2x2 + 4x3 + 8x5). 



46 DIFFERENTIAL CALCULUS 

4. y ^{x-\- 1)5 {2x- 1)3. dy = (16a; + 1) {x + 1)4(2^ - lydx. 

e / 1 \ . / dy a — 3 X 

5. , = (a+x)Va-:,. i = 2vj3^- 

6. 2/ = (l-3a;2+6x4) (1+^2)3. d^/ = 60 a;^ (1 + ^2)2 da;. 

7. ^ = (.^-a^)\ dj^ ^{x^-a^y 

dx ^^l 

- _ g; + g^ dy _ b — a^ 

^~x + 6* rf^~ (re + 6)2* 

^ (x + b) d (x + o2) - (x + a')^ {x + 6) 

10. y =x {x^ + 5)^. 
2rc4 



11. y 

a'- — x" 

12. ?/ = Vcfx2 + 6a: + c. 



14. 2/ = 

15. ?/ = 



Va2 - x2 



(a; + 6)2 


d?/ 


2x 


dx 


{X - 1)3 


dy _ 

dx 


: 5 (^3 + 1) (^3 + 5)J. 


dy 


8 a^x^ - 4 x5 


dx~ 


(a2 - x2)2 


dy _ 


2ax + h 


dx 


^ Vaa;2 +6a; + c 


dy 


1 


dx 


(1 -x)Vl- x2 


dy 


a2 


dx 


V(a2-a;2)3 


dy 


na;"~i 


dx 


(1 + x)'H-i 


dx 
dy. 


2 na:"-i 
(X- - 1)2 



(i+xr 

Va2 + a:2 - X dx a? LVa? + a;2 J 

Rationalize the denominator before differentiating. 

n—\ 7 w— 3 

18. a; = « (f2 + a?)~^ . , ^ = (n^2 4. ^2) (^2 _|. ^2) "2". 

19. A vessel is sailing due north 20 miles per hour. Another vessel, 
40 miles north of the first, is sailing due east 15 miles per hour. At 
what rate are they approaching each other after one hour? After 2 
hours? Ans. Approaching 7 mi, per hr.; separating 15 mi. per hr. 
When will they cease to approach each other, and what is then their 
distance apart? Ans. After 1 hr. 16 m_. 48 sec; 24 mi. 

20. If a body moves so that s = Vi, show that the acceleration is 
negative and proportional to the cube of the velocity. Negative sign 
shows what ? 



FORMULAS AND RULES FOR DIFFERENTIATION 47 

21. Ji X = at and y - ht — -^cf, find -^ and -t- 



2 ' dx dy 

dx _ dx / dy _ a 
dy dt I dt b — ( 
bx _lcx^ ^ . dy _b _cx _b cat _ b — ct 
a 2 a^ ' " dx~ a a^ a a^ a 



dy _dy ldx_bj-ct^ dx_dx ld^_ _a_^ rRv Art IPfoU 

di'Tt/ dt~ a dy- dt/ dt'b-ct ^^^ ^^' ^^^""^'^ 

Or y = 



22. If p = v^ and = ^2 - 10, find ^. 

ua 

^^ = ^/§ = ^/2^ = — 3- (By Art. 19, Remarks.) 
dd dt / dt 2t^/ 4:t^ 

Or p2 = ^ = (0 + lO)^ .-. 2pdp= 1 = ~ 

2(9 + 10)* 4t^ 

23. The equation pV = C expresses Boyle's law, C being a constant. 

Find Ig: and ^- (See Ex. 5, Art. 20.) 
dV dp 

24. The heat H required to raise a unit weight of water from 0° C. 
to a temperature T is given by the equation, 

ff = r + 0.00002 T2 + 0.0000003 T3. 

(a) Find -ttf- (&) Compute the numerical value of the rate for 
T = 35°. Ans. (b) 1.0025025. 

25. A vessel in the form of an inverted circular cone of semi-vertical 
angle 30°, is being filled with water at the uniform rate of one cubic foot 
per minute. At what rate is the surface of the water rising when the 
depth is 6 inches? When 1 foot? When 2 feet? 

Ans. 0.76 in.; 0.19 in.; 0.05 in., per sec. 

26. Show that the slope of the tangent to the curve y = x^ -{- 4: is 
never negative. Find the slope for x = 0, ior x = 2. For what values 
of x does the slope decrease as x increases ? 

LOGARITHMIC AND EXPONENTIAL FUNCTIONSo 
32. Formulas and Rules for Differentiation. — 
[VIIIo] d (logbx) = — dx (x positive), 

X 



(m = \ogbe = 0.434 . . . , for 6 = 10). 

x) = - dx {x positive). 

(m = loge6 = l,e = 2.718 . . . ). 



[Vlllb] d (loge x) = - dx {x positive). 

X 



48 DIFFERENTIAL CALCULUS 

The differential of the logarithm of a variable is the product 
of the modulus of the system and the reciprocal of the variable 
by the differential of the variable. 

[IXJ d (5*') = b'^ loge h dx (b positive). 

[IXfc] d{e^) = e''dx. 

The differential of an exponential function with constant base 
and variable exponent is the product of the function and the 
Napierian logarithm of the base by the differential of the 
exponent. 
[X] d (y^) = y- logeydx +*y^^ dy, 

(y positive and independent oi x). 

• The differential of an exponential function with base and 
exponent variable is the sum of the results obtained by differen- 
tiating as though the base were constant and then as though the 
exponent were constant. 

33. Derivation of [VIIIJ and [Vlllfi].— 

(i) Taking n an arbitrary constant, let 

X = ny. (1) 

logb X = logb {ny) = logb y + log6 n. (2) 

d (logt x) = d (log5 y)[+d (logj, n) = 0]. (3) 

Differentiating (1) and dividing result by (1), 
dx _dy 
X ~ y' 

Dividing (3) by (4) gives as result, 

d{\og,x)l^^ = d{\og,y)l^'- (5) 

It is manifest that the equal ratios in (5) are constant for 
any particular value of y. Let m denote the constant value 
of the ratio when y = yi; then 

d (logb x) = ~ dx, (6) 

X 



(4) 



DERIVATION OF [Villa] AND [VIIIj,] 49 

when X = nyi] and, as n is- an arbitrary constant, nyi denotes 
any positive number. Hence (6) or [Villa] holds true for 
all positive values of x, m being a constant. The constant 
m is called the modulus of the system of logarithms, whose 
base is denoted by b in this derivation. The general base is 
often denoted by a. 

The system whose mQdulus^.is unity is called the Napierian 
or natural system. The symbol for the base of this system 
is e, called the Napierian base from the name of the dis- 
coverer of logarithms. 

Hence d (loge x) = - dx, [VIIIj] 

X 

(ii) By the general method of limits, {x positive.) Let 
y = log6 X. y + ^y = logt (x + Ax), 

A2/ = log6 {x + Ax) - \ogbX = \ogb (1 + -~] , 

X 

Raising each member to -r— power gives 

X 

x^ I Ar\^ 

6-=(l+^). 
The limit of each member as Ax = gives 

X 

lim6 ^^ = limH =lim 1 + - , puttmg — ■=--, 

SO that (if X is not zero), as Ax = 0, n = 00 ; 

lim 6 ^^ = 6 ^^ = lim 1 + - = e 

Az=0 n=oo \ nj 

(denoting the hmit by that letter) ; 

dx ■ dx X X 

(m = log6 e = the modulus) ; 



50 DIFFERENTIAL CALCULUS 

/. dy = d (logb x) = — dx. [Villa] 
Hence, 
d (loge x) = - dx, since loge e = 1 = the modulus. [VIIIs] 

X 

The limit of ( 1 + - ) as n is increased without Hmit is e, 
the Napierian base. (See next Art. 34.) 

34. Lim (l + -T = e. (See Ex. 7, Art. 221.) 

n=ai \ TlJ 

When n is a positive integer, by the Binomial Theorem, 



(■+3"= 



, 1 n{n-l) 1 n{n-l){n-2) \_ 
"^ n"^ 1-2 'n^"^ 1.2.3 \^^ 

\ n) . \ n)\ nl 



=w+-^^:^+ ^.-g • + •••• 

In the expansion there are (n + 1) terms in all, and every 
term after the second can be written in the form given to the 

3rd and 4th terms. As n = oo , - = 0, 

n 

■■■ ^„(i+3"=i+i+l+A+2-.i4+--- 

= e = 2.7182818 .... 

Note 1. — The limit is denoted by e, which is an irrational 
number, and was proved by Hermite, in 1874, to be trans- 
cendental or non-algebraic. The number e was the first 
number to be proved transcendental. Not until 1882 was 
the attempt to prove the number w transcendental successful. 
This was finally done by Lindemann. The proofs consist in 
showing that neither of the two numbers is the root of an 
algebraic equation with integers for coefficients. Algebraic 
real numbers are defined as those real numbers which are 
roots of such an equation. The importance of these two 
numbers, considered the most important in mathematics, 



DERIVATION OF VALUE OF e 51 

warrants gDme notice. They are connected by the remark- 
able relation, e^^^^ = 1. (See Ex. 10, Exercise XLIII.) 

Note 2. — The above derivation of the Umit is not com- 
plete, for the result is true not only when n is ''a positive 
integer," but also for n positive or negative, integral, frac- 
tional, or incommensurable. The value of the hmit, e, can 
be easily computed to any desired degree of precision by 
taking a sufficient number of terms of the series. Twelve 
l^erms gives the result correct to seven decimals; that is, 
e = 2.7182818 . . . 

By comparing the sum of (n + 1) terms of the series with 

the sum, 1 + 1 + - + ^+ • • • 7^1 , which is greater than 

the other, and equal to 3 — i''~S it is manifest that no matter 
how great n may be, the sum of the (n + 1) terms is certainly 
finite and less than 3. The 

may be considered as 5, or as usually written, 

to infinity. (See Ex. 5, Art. 215.) 

Without expanding, the lim f 1 + - ) can be computed to 

any desired number of decimals by giving increasing values 
to n; thus, 

(1 + ,1^)10 = 2.59374. 

(1.01)100 = 2.70481. 
(1.001)1000 = 2.71692. 

(1.000001)1000 000 = 2.71828. 

The last number agrees with the value of e, the required 

limit, to five decimals. 

1 

Corollary. — Lim (1 + n)" = e. (See Ex. 8, Art. 221.) 

n=0 



52 DIFFERENTIAL CALCULUS 

35. Derivation of [LXa] and [IX^]. — 
(i) Let y = h^, then lege y = x lege b. 

d (lege y) = d{x lege 6) , or — = lege h dx, 

:. dy = d (6^) = h'^logehdx (b being positive). [IXa] 
Hence, d(e^) = e'^dx (since logeC = 1). [IX^] 

(ii) li y = h"", X = logb y. dx = d (\ogb y) = —^ — ^* 
.*. dy = , dx = h^'logehdx (since -. = loge^J. [IXa] 

Hence, c^(e^) = e^do;. [IX&] 

(iii) By the general method of Hmits. Let 

y = G^. y -\- Ay = e^+^^. 
Ay = e^+^^ — e^ = e^ (e^^ — 1). 

A^ _ e^ (e^^ - 1) 

Ao; ~ Ax 

lim r^l =p= lim r.= f-^1)] = .^ lim f^^) 

A.=oL^:cJ dx A.=oL V Ao; /J Az=0\ AiC / 

= e^f since Urn (^-r— -)= l) (C^or., Art. 36); 

/. dy = die'') = e^dx. [IX^] 
Corollary. — d (h^) = ¥ loge h dx. [IXa] 

1 XI 



For if — = loge h, ¥ = e^ , since 6 = e^ ; 






m m 



:. d (6^) = ¥ loge h dx. 
36. Limfl + -V = e^ 



DERIVATION OF [X] 53 

1 + - , if X 7^ 0, by putting n = Nx, when 
nj 

n = 00 so is AT" = 00 ; hence 

(■+3"=('+r=i('4)T. 

and 

!f-(^ +3" -Mi' +W\ = hi' +^)7 = '-' 

since lim/ (a;) = / (lim x), the case of a function of a function. 
(See Remarks, Art. 19.) 

By exactly the same method as in Art. 34, it may be 
shown that 

(x\^ 
1 + - j for positive integral values of n. 

It can be shown that the limit of this series is a finite 
number for all finite values of x no matter how great n may 
be. (See Art. 213 and Ex. 5, Art. 215.) 

Corollary. — Limf j = 1, which may be put in the 

form, lim ( — r ) = 1. 

Ax=o V Ax / 

37. Derivation of [X]. — 

Let z = y"", then loge z = x loge y. (y Dositive and inde- 
pendent of X.) 

d{logeZ) = d{x\ogey)i 

~ = \ogeydx-\-x-^; 

:. dz = d {y") = y^ loge ydx + xy"-^ dy (y positive). [X] 

Note. — Formulas [VII], [IXa], [IXb] are seen to result 
from [X] as special cases. 



54 DIFFERENTIAL CALCULUS 

Let y = x"", then loge?/ = nlogeX, 

d{\ogey) = d{n\ogex), 
dy dx 

y X 

:. dy = d (x^) = nx''-^ dx. [VII] 

If X were negative, to avoid logarithms of negative num- 
bers, both members of y = x"" are squared before differ- 
entiating. 

This derivation of [VII] includes the case where n is 
incommensurable. 

38. Modulus. — In Art. 33 (ii), it appears that logb e is 
the modulus of the system of logarithms whose base is h. 
Hence, when the base is 10, as in the common system, and 
the value of e is known, a table of logarithms will give the 
value of the modulus of the common system to as many 
decimals as the table gives. The modulus of the common 
system, denoted by M, is logio e = 0.43429 ... If this 
value of M is deduced independently of any knowledge of 
the value of e, which can be done ; then the value of e can be 
gotten from a table of logarithms ; for, since M = logio e, 
then e = 10^; that is, e is the number whose common 
logarithm is 0.43429. . . . 

In Art. 35 (i) and (ii), it appears that loge 6 is equal to 

^ log. 10 = r-^„ = -rj^ = 2.3026 



log6 5' *• ^V" logio e .434 

approximately. (See Ex. 6, Art. 215.) 

To get these results independently, let x be any number 
whose logarithm in the system with base 10 is I, and in that 
with base eisV; then 10^ = x and e^' = x; 

/. 10' = e^'. (1) 

Let 10^ = e; (2) 

/. 10^ = 10^^'; .-. l=MVoY \, = M, (3) 



MODULUS 55 

and since 10 and e are constant, so also is M, From (2), 
M = logioe, or from il) I = logioe^' = Tlogioe; 

J, = logioe = M] or in general, log6e = m. 

Since I = MV, or logio x = M loge x, it follows that the com- 
mon logarithm of any number is equal to M times the 
Napierian logarithm of that number. 

Now d (logio x) = Md (loge x) or ^Jj^^''^^ = M, 

d{\0geX) 

or m for base h, and dividing [Villa] by [Vlllb] gives 
jn i = ^) the modulus; 

d (loge X) 

:. M = logic e = modulus of common system (h = 10), 
and m = loge e = 1 , modulus of natural system {b = e). 

From (1) above, I log^ 10 = ^ or r- = - — — - = M, by (3) ; 

approximately. 

To summarize m two equations: 

Common log = 0.434 times natural log. 
Natural log = 2.3026 times common log. 

Note. — Since the modulus of the natural system is unity 
the differentials of logarithms are simpler when the logarithms 
are in that sj/stem; hence, in the Calculus and in most 
analytic work, Napierian logarithms are employed for the 
most part. Any finite number except one could be made 
the base of a system of logarithms. For computation the 
common logarithms are the best, as having the base 10 
affords rules for the integral part of the logarithms and 
obviates the necessity of that part appearing in the tables. 
It is usual in writing log for logarithm to omit the subscript 



56 DIFFERENTIAL CALCULUS 

indicating the base, when no ambiguity results. Hereafter, 
when no subscript to log appears, e will be understood. 

39. Logarithmic Differentiation. — Exponential func- 
tions and also those involving products and quotients are 
often more easily differentiated by first taking logarithms. 
This method which is used in the last two derivations (Art. 
35 and Art. 37) is called logarithmic differentiation. 

To derive [Va], let z = uy, then log z = log u + log y, 

d(log2) = ^ = 1* + ^ = d{\ogu)+d(\ogy); 
z a y 

:. dz = d (uy) =' ydu -\-u dy. 

To derive [Yb], let 

V = uyz, then log V = log u -\- logy + log z, 

dV du . dy . dz 
d(logF) = -^ = - + -+-; 

.*. dV = d (yz) = yzdu -{- uz dy + uy dz. 
To derive [VI], let z = y/x, then log ;s = log 2/ — log x, 

d{logz) =-^= ^-^ = d{logy) -dilogx); 

z y J. 

• dz = d(-]= — - — = ^dy - ydx 
\x/ X x^ x^ 

40. Relative Rate. Percentage Rate. — The logarithmic 
derivative of a function may be defined as the relative rate 
of increase of the function. Thus, when y = f (x), 

dy 

— = \, / is the relative rate of y. 
y /W 

Hence, when z = xy and therefore, log z = log a; + log !/; 

dz dx dy 

dx dx , dx. 

z ~ X y ' 



EXERCISE 

that is, the relative rate of increase of a product is the sum of 
the relative rates of increase of the factors. If the logarithmic 
derivative is multiphed 100 times, the product expresses the 
percentage rate of increase. Thus when 

^ = Ky, 100— =100X 
ax y 

is the percentage rate of increase, and is here constant. 



EXERCISE IV. 

By one or more of the formulas [I] to [X], differentiate:* 

1. y= logbx^ = Slogbx. , -£ = -^ = —- 

n n \-i dy 31ogioe,i „ 1.302... ,, .„ 

2. y = (logio x)\ ^ = — 1~ (logiox)2 = (logiox)2. 

Z. y = xlogx. -^ = log a; + 1. 

4^ 1 , dy ^ 1 + log a; 

xlogx dx (a;logx)2 ' 

*• y-^os''^^ = \oe(ax-b}-\os(ax+b). | = ^|^. 
6. 2/ = log ^+ - = log (1 + VJ) - log (1 - Vi). 

7.. = iogWr^. 1 = 2^. 

B. y = 6^e^ ^ = (1 + log h) 6==e^. 



9. y =log(a^ + 6=^). 



dy _ g^ log a + b^ log b 
dx~ a^ + 6^ 



10. 2/ = x^ 5^. ^ = x'b' (5 + x log 5). 

11. y =x^ ^ = a;^(loga: + l). 

ax X 

* In some of the examples logarithmic differentiation is employed to 
advantage; that is, take logarithms first and then differentiate. 



58 DIFFERENTIAL CALCULUS 

Here log y = e^ log x :. — = e^ — + log xe^ dx. 

y ^ 

13. y = x'og ^. dy = 2 x^og ^"^ log a; • da;. 

U. , = log(logx). | = ,t4^- 

16. a;^°sa = a}ogx^ (Differentiate]^both members and verify results.) 

16. {x + e^)4 = a:^ + 4 a;%^ + 6 a:2e2^+ 4 xe^"" + e*^. (Do as in 15.) 

17. (e^ + e-^)' = e2^ + 2 + e-2^. (Do as in 15.) 
c^ — e-^ dy 4 



18. 2/ 



e^ + e-^ dx (e^ + e-^)2 

e^ dw 1 



19. 2/ = log,^^. ^^ j^^, 

20. y = (log x)^. ^1 = (log xr i-^ + log log x\ - 

21. Find the slope of the curve y = logio a;, or a; = 10^, showing that 
the results are identical. What is the value of the slope at (1, 0)? 
What is the slope of the curve y = logg x, or x = e^, and its value at 
(1,0)? 

2£. Find the slope of the curve x = loge y, or y = e^; and note that 
the slope at any point has the value of the ordinate at that point. Value 
of the slope ata:=0? Ata: = l? Ata;=— oo? 

(X _x\ 

ea + e a/ at a; = 0. 

Ans. 0. 
What is the abscissa of the point where the curve is inclined 45° to 
the a:-axis ? 

Ans. X = a loge (l + ^2). 
24.* Find the value of x when logio x increases at the same rate as x. 

Ans. X = logio e = 0.4343 . . , 

dx 
* Since d (logio x) = logio e • ^- ; 

dx = x' "^f""^^"^^ = 2.3026 X . d (logio x) ; 
logio e 

hence, any number N increases about 2.3 AT times as fast as logio A^. 
When 

N = 0.4343 . . . ,dN = 0.4343 X 2.3026 d (logio AT) = d (logio iV). 

Find how much faster x is increasing than logio a; for x = 1. 

Ans. 2.3026 x = 2.3026 . . . 



RELATIVE ERROR 59 

25. When the space passed over by a moving point is given by 
s = ae* -{- be~^, find the velocity and the acceleration, showing that the 
acceleration is equal to the space. 

26. Find the slope of the curve y 

27. Find the slope of the curve y = 

e^ -\- e <^ 

28. Find the derivative of the implicit function y in e^^ = xy. 
Passing to logarithms : 



= -{ea- 


■ e 


"«/ at 


X = 


0. 




X 

ea — e 

X 


X 

a 

X 


at X ■- 


= 0. 


Ans. 
Ans. 


1, 

1 

a 



x + y =\ogx + \ogy. -^ = 
Passing to log 
ylogx = X log y 



dy y{l- x) 



dx x{y — 1) 

dy 2/2 _ j.y log y 



dx x^ — xy log x 
30. Find the slope of the probabihty curve y = e-^"-. 

Ana. —2xe-^^. 
What is the value of the slope at x == ? 

Ans. 0. 
2 

At X = 1? Ans. 

e 

41. Relative Error. — Since when y = fix), the relative 
rate of increase of y is 

^ df{x) 

dx _ dx _ f {x) 

y ~1W~W)' 

where dy = f (x) dx ; 

hence, Aiy = / (x) Ax, (1) 

and ^ ^=^#Ja., .(2) 

y fix) 

are approximate relations. The relation (1) is useful in 

finding the error in the result of a computation due to a small 

error in the observed data upon which the computation is 

based. The relation (2) gives approximately the relative 

Ay 
error — ^• 

y 



60 DIFFERENTIAL CALCULUS 

1. Thus, to find an expression for the relative error in the 
volume of a sphere calculated from a measurement of the 
diameter when there is an error in the measurement. Here 






AD ^gAD 



7rD» D 

Hence, an error of one per cent in the measurement of the 
diameter gives approximately an error of three per cent in 
the calculated volume. 

2. Again, from the formula for kinetic energy X = i mv^, 
to show that a small change in v involves approximately 
twice as great a relative change in K. Here 

AK V Av ^Av 
-^^ = m-i ^ = 2 — 

3. If a square is laid out 100 ft. on a side and the tape is 
0.01 ft. too long, an errojof y^iy of one per cent, the relative 
error in the area is, approximately, 

M - 2a: ^ - 200-^:^ - 0002 

or if of one per cent. 

1 +-) 

= e=^, in Art. 36, arises in a variety of problems. When a 
function has the general form 

y = ae^, (1) 

then -p = hae^'' = hy; 

that is, the rate of change of the function is proportional to 
the function itself. Many of the changes that occur in 
nature are in accordance with this law, called by Lord Kelvin 
the compound interest law. It is so called because of the fact 
that the amount of a sum of money at compound interest 
has a rate of change at any value proportional to that value, 
when the change is continuous. 



THE COMPOUND INTEREST LAW 61 

Let A = amount, r = rate per cent, P = principal, and t 
number of years; then, 

A = P(l + r)\ when interest is compounded yearly; 
1 + -1 , at n equal intervals each year; 

. A = limP ["[l ^-VT = Pe'i (by Art. 36) (2) 

is the amount when the interest is compounded continuously. 

dA 
Here A = Pe*"' and —rr = rPe'"' = rA, hence, the rate of 
dt 

change of the A is proportional to the value of A, the factor 

of proportionality being the rate per cent at which the 

interest is reckoned. As a comparison, it may be noted that 

$L00 will amount to $2,594, in ten years with interest at 10 

per cent, compounded yearly; while the amount will be 

$2,718 when compounded continuously. 

If in A = Pe'"', t increase in any arithmetical progression, 
whose common difference is h, A will increase in a geometrical 
progression whose common ratio is e''^; for if t become t + h, 
A will become Pe'"('+^), that is, Ae""^. Hence A is a quantity 
which is equally multiplied in equal times. 

The density of the air towards the sea level from an eleva- 
tion is a quantity which is equally multiplied in equal 
distances of descent, for the increase in density per foot of 
descent is due to the weight of that layer of air which is itself 
proportional to the density. Many other instances occur 
in physics. 

When bacteria grow freely the increase per second in the 
number in a cubic inch of culture is proportional to the 
number present. The relation between the number N and 
the time t is expressed by the equation, 

dN 

N = Ce^'; .-. ^ = kCe'' = kN, (3) 



62 DIFFERENTIAL CALCULUS 

where N is the number of thousand per cubic inch, and k 
is the rate of increase shown by a colony of one thousand per 
cubic inch. So many instances of this kind are found in 
organic growth — where the rate of growth grows as the 
total grows — that the law is called the law of organic growth, 
as well as the compound interest law. 

When a quantity has a rate of change which 'is proportional 
to the quantity itself, if the functional relation is expressed 
by an equation, it must be of the form (1). 

In the case of the density of the air, the relation (see Art. 
226) between the density p and the height h above the sea 
level is expressed by 

where po is the density at the sea level and A; is a constant to 
be determined by experiment. From barometric observa- 
tions at different altitudes, it has been found that at the 
height of 3| miles above the earth's surface, the air is about 
one-half as dense as it is at the surface. Hence, to deter- 
mine k, 

£ _ .-3.5;t _ 1 ; 
Po"' ~2' 

.-. -3.5 fc = log 0.5, or A; = ^^^ = 0.198; 

.*. p = poe~^-^^^^, where h is in miles. (5) 

Here, as h increases in arithmetical progression, p decreases 
in geometrical progression, the force of gravity and the 
temperature being taken constant. The varying density at 
different heights is found by giving values to h; thus, making 

h = 35, gives — = 0.001 ; hence, according to this law at the 

Po 
height of 35 miles the density of the air is about one-thou- 
sandth of the density at the sea level. As the pressure p is 



THE COMPOUND INTEREST LAW 63 

proportional to the density, p = k'p; and the same law holds 
for the pressure of the air; hence, 

p = poe-^'^ ••• ^ = -kpoe-^'^ = -k'p^ (6) 

where /c' is a constant to be found by experiment. 

Knowing the pressure at the sea level and observing the 
pressure at some height, ¥ is determined; or it can be 
determined from the value of the pressure at any two differing 
heights. When the pressure is expressed in inches of mercury 
in a barometer, the pressure in lbs. per square inch = 0.4908 
X barometer reading in inches. Taking pq = 30'' when 
/i = 0, and p = 24'', say, when h = 5830 ft., /b' is readily 
computed. In millimeters the equation is p = 760 g-'^/sooo, 
where h is in meters. 

The relation between the decomposition of radium and 
time is expressed by the equation 

q = qoe-^'; .*. J= -/bgoe"*' = -kq, (7) 

where ^o is the original quantity and q is the quantity remain- 
ing after a time t. The constant k can be found from the 
fact that half the original quantity disappears in 1800 years. 
The relation between the varying difference of tempera- 
ture of a body and that of the surrounding medium and the 
time of cooling is expressed, according to Newton's Law, by 

8 = 6o6-^'; .-. ^ = -kdoe-'^ = -k8, (8) 

where 5 = r — to, the difference in the temperature of the 
body and that of the medium, 5o = ri — to, the difference 
when t = 0, k a constant; that is, t = to + (ti — to) e~^\ 
where —kt indicates the body is cooling. 



64 . DIFFERENTIAL CALCULUS 

TRIGONOMETRIC FUNCTIONS. 

43. Circular or Radian Measure. — The formulas for 
differentiation of trigonometric functions are simpler when 
the angle is measured in radians than in degrees. Hence, 
in the formulas that follow, the angle will be in radians. 

A radian, the unit of circular measure, is an angle which 
when placed at the center of a circle intercepts an arc equal 
in length to the radius. 

180° 
Since 2 irr is arc of 360°, a radian equals , or 57.3° 

TV 

nearly. In circular or radian measure, an angle in radians 
is equal to the length of the intercepted arc divided by the 

radius; Q ='-, where d is angle in radians, s is number of 

units in arc, and r is the number of units in radius. Hence, 
s = rd; that is, in any circle the length of an arc equals the 
product of the measure of its subtended central angle in 
radians and the length of the radius. If r = 1, then s = d; 
that is, the arc and the angle have the same numerical 
measure. Trigonometric functions are called circular func- 
tions. 

44. Formulas and Rules for Differentiation. — 

[XI] d* sin e = cos d0. 

The differential of the sine of an angle is the cosine of the 
angle by the differential of the angle. 

[XII] d cos e = - sin e d0. 

The differential of the cosine of an angle is minus the sine of 
the angle by the differential of the angle. 

[XIII] cltan0 = sec^0d0. 

The differential of the tangent of an angle is the secant 
squared of the angle by the differential of the angle. 

[XIV] ' dcot = - cosec^ dQ, 

* Parenthesis after d omitted when no ambiguity results. 



DERIVATION OF [XI] AND [XII] 



65 



The differential of the cotangent of an angle is minus the 
cosecant squared of the angle by the differential of the angle. 

[XV] d sec e = sec 6 tan dS. 

The differential of the secant of an angle is the secant of the 
angle by the tangent of the angle by the differential of the angle. 

[XVI] d cosec 6 = - cosec cot d0. 

The differential of the cosecant of an angle is minus the cose- 
cant of the angle by the cotangent of the angle by the differential 
of the angle. 

45. Derivation of [XI] and [XII]. — 

I. Let the point P{x, y) move along the arc XPY of a unit 
circle. Denote the number of linear units in the arc XP by 
s, and the number of radians in angle XOP by 0. 

Then B = s, y = dnS, x = co^d; 

:. do — dsj dy — d sin 6, dx = d cos d. 
Angle DTP equals 6, and dx is nega- 
tive; hence, from the triangle DTP^ 
by (d) Art. 10, 

dy = dsind = cos Odd, since ds = dd; 

dx = d cos = — sin ^ dd, 

since —dx = sin 6 ds and ds = dd. 

It is seen that dy and dx in the figure 
are what the changes of the sine and 
cosine of 6 would be if, at the value XOP of 6, the changes 
were to become uniform. 

II. By the general method of limits. 

Let y = sin 6, then y + ^y = sin {$ + AS); 




Ay = sin {d + Ad) — sin d 

AS 

ein 

Ay 



2 sin 



sm 



Ad 



Ad 
2 



cos 



hf) 



AS I 
-cos(^i 

/ 



+ 



A^^ 



A^^ 



As 



A^ 
2 



= 1, 



, by Trig.; 

[(Art. 46). 
as AQ = 0; 



66 



DIFFERENTIAL CALCULUS 



de aJ?oLa^J 



lim 

A9=0| 



Sin -2-1 



2 



j-Kf) 



cos^; 



/. dy = d sin 6 = cos Odd. 
Corollary. — d covers 6 = d{l — sind) = — cos 6 dO. 

Now let a; = cos ^ = sin (S — ^) i 

dx = dcosd = dsinl^ — dj = cosf^ — djdl^ — d); 

.*. dcosd = — sin d dd. 
Corollary. — d vers S = d{l — cos 6) — sin Q dB. 

Let d be the number of radians in the angle NO A, where 
the angle is taken acute; by Ge- 
ometry, if AT and BT are tan- 
gents at A and B, 
then, 

chord AB < arc A5 < AT + BT, 
and therefore 

MA < arc iVA < AT. 
Hence 




that is, 

dividing by sin 0, 



MA arc iVA AT 
OA^ OA ^OA'' 

smB <B < tan0; 



1<J_<J_ or l>EEf>eose. 

sm0 COS0 

Thus the ratio -— r— hes between 1 and cos B. 



LIMIT OF RATIO OF INFINITESIMALS 67 

When 6 approaches as its hmit, cos d approaches 1 as its 
hmit; therefore, also sin ^/0 approaches 1 as its hmit. (See 
Art. 215, Ex. 1.) 

Corollary, — Since hm — -— = 1, 

6=0 ^ 

,. 2 MA ,. chord AB , /a ^ or. i x 
lim ^ ^r A = lim 7-75- = 1. (Art. 22, also.) 

It may be noted that, 

since -rjF = 1 — n = cos B, when = and cos 0=1, 
AT tan^ ' ' 

sin0 and tan 6 approach equality; and, since the arc B is 

intermediate in value between sin B and tan B, the three 

functions approach equality as the angle B nears zero. So 



Hm (7 — ^1= 1 and lim \—z — )= 1, as well as hm (^^—] 



1. 



These are fundamental examples of the ratio of infini- 
tesimals approaching a constant value as a hmit. Con- 
sider again the equality of ratios, -j^ = -y^-- Suppose the 

points A and B approach N; so long as A and B are not 
coincident, that is, so long as AB is really a chord, the 
equahty still exists. The ratio MA : AT may be considered 
a function of OM, or equally well a function of the angle 
NO A. As OM approaches ON as its limit, or as the angle 
NO A approaches zero as a hmit, the ratio MA : AT ap- 
proaches 1 as its hmit. The nearer OM gets to ON, or the 
nearer A gets to N, the nearer does the ratio MA : AT get 
to unity. The crucial fact is that the reasoning is vitiated 
if OM becomes actually equal to ON; for then the triangles 
will cease to exist, the terms of the one ratio will be zero and 
those of the other will be identical, and the equation on 
which the reasoning is based could not be estabUshed. 



68 DIFFERENTIAL CALCULUS 



47. Derivation of [XIII]. — 

Since tan^ = -, atan^ 

cos^ 



\cos0/' 



, , ^ COS ^ d sin — sin ^ c^ cos d 
a tan 6 = 



cos^^ 
(cos2^ + sin2 0)d^ 



sec^ d de. 



cos^^ 
48. Derivation of [XIV]. — 

Since cot 6 = tan ( 9 — ^ ) ; 

.-. dcot^ = dtQ.n(^-d\ = sec^^^-^^d^^-^] 



= — cosec^^d^. 
49. Derivation of [XV]. — 

Since sec0 = 

.*. d sec 6 



cos^' 

sin d do 



Vcos^/ 



cos^^ 
= sec^tan^c?^. 

50. Derivation of [XVI]. — 

Since cosec ^ = sec (^ — ^j ; 

.*. dcosec0 = dsec(:^ — 6] = secf^ — ^Jtan (^ — djdl^ — dj 
= — cosec 6 cot 6 dd. 

Note. — In the derivations of the formulas for the cosine, 
cotangent, and cosecant, as given, it may be noted that, as 
in the last example of Art. 31, the formula for the derivative 
of the function of a function has appropriate application. 

Thus for COS0, let a; = cos0 = sinf- — ^j and <!> = -x — dy 



NOTE ON [XI] 69 



dx _ dx d4 
*^®"' dd~"d4>"dJd' 



^^ de 



r^^^ =i^^^^ ^^^(i- ^)= - ^^^^ = - ^^^(l- ^)' 



-T^ COS = — sin ^ or d cos ^ = — sin c?0. 



In practice the actual substitution of the auxiliary symbol <t> 
may be dispensed with. 

The formula applies to such functions as 2/ = sin {ax + 6). 
Thus put z = ax -\-h, making y = sin 2 ; then 

dy _dy dz 
dx dz dx 

or J- sin (ax + 6) = -r- sin z X -^ (ax -\- b) = cosz^a; 

.*. -r- sin (aa; + 6) = a cos (ax + 6) 
or d sin (ax + 6) = a cos (ax + 6) dx. 

Again, let the function he y = sin^ (ax + 6) and put z = 
sin (ax 4- h) ; then 

= 2 2; X a cos (ax + 6) 
= 2 a sin (ax + 6) cos (ax + 6) ; 
.*. d sin^ (ax + 6) = 2 a sin (ax + 6) cos (ax + b) dx, 

51. Note on [XI]. — If the angle is measured in degrees, 

then d sin = -^ cos 6 dd, since 6 degrees is r-^ radians 
loU loU 

and sin(9° = sin^^j; 

.*. dsin0° = dsinf^^J 



70 DIFFERENTIAL CALCULUS 

It is thus seen that the formulas for differentiation of the 
trigonometric functions are simpler when the angle is 
measured in radians than when measured in degrees. For 
the same reason that Napierian or natural logarithms are 
employed in differentiation, radian or circular measure is 
used for angles of the trigonometric functions, when differ- 
entiation is to be done. 

62. Remarks on [XI]. — The fundamental Hmit of Art. 

46, lim ( -— — - ) = 1 means that when is a small number 
e=o\ o I 

sin Q is approximately equal to Q. For angle of 1°, ^ = -x^r 

= 0.0174533 . . . , sin6' = 0.0174524 . . . ; so they are 
equal to five decimals. Of course for angle of 1' or 1'', they 
are equal to a great many more decimals, but they are never 
exactly equal however small the angle may be, since the sine 
is always less than the arc. 

Since c? sin ^ = cos B dd, if the value 0° is taken for 6 and 



(=^)ford^(=A^), 



dsmd = cosO°-^ = .0174533 ... or ^^ = 1, 
180 dd 

A sin (9 = sin (0° + Ad) - sin 0° = .0174524 . . . = sin 1^; 

A sin _ .0174524 ♦ ♦ » ^ . y A sin6> _ casing _ 1 

Ad ".0174533 . . . ' M% Ad ~ dd ~ J' 

e=o° 

From ,. = cosd= cosO° = 1, it is seen that, at d = 0°, 
da 

the sine of d is changing at the same rate as d is changing; so 

the slope of the curve y = sind is unity at the origin, and the 

tangent to the curve at that point makes an angle of 45° with 

^-axis. The conditions are the same at ^ = 2 tt. As — rr— 

dd 

IT 

= cosd = COS 90° = 0, at = ^, the sine of d is not changing, 



REMARKS ON [XI] 



71 



the rate being zero, and the tangent to the curve at that 
point is parallel to ^-axis. At ^ = tt, cos 180° = —1, so the 
sine of 6 is decreasing, at that value of 6, at the same rate as 
6 is increasing, and the tangent to the curve at that point 
makes angle of 135° with ^-axis. Thus the rate of change of 
the sine of 6, at any value of 6, can be found ; and the differen- 




7/ = sin 9 



tial of the sine, the change if the change became uniform, 
will always differ from the increment, the actual change of 
the sine, when the angle is given an increment. In taking 
sines or other functions from tables by interpolation, the 
changes are assumed as uniform within allowable limits of 
error. 





EXERCISE V. 


1. y = sin x^. 

2. y = sin2 x. 
Z. y = cos ax. 




dy ^ 2x cos x2 dx. 

dy =^ 2 sin x cos xdx = sin 2 x dx. 

dy = — a sin ax dx. 


4. y =f(d) = tan^ d. 




^ =f(0) =m tan"*-! sec^ 0. 


5. f(d) = tan 30 + sec 

6. fix) = sin (logaa;). 

7. f{x) = log (sin ax). 
Q sin a: + cosx 


3 0. 


f id) = 3 sec2 3 + 3 sec 3 tan 3 0. 
/' ix) = l/xcos (log ax). 
/' ix) = a cot ax. 
dy 2 sin x 


8. y g. 


dx e^ 


9. f(d) = log (tan 00). 

10. fid) = log (cot 00). 

11. fid) = tan (log 0). 




f'(o)-- ^"^ - ^^ 


•' ^"' 2 sin (o0) cos (o0) sin(2o0) 
/'(0) = -2 a/sin (2 00). 
fid) =l/0secMlog0). 
,, ,^. sec tan , ^ 



12. /(0) = log (sec 0). 



fid) 



sec0 



= tan 0. 



72 DIFFERENTIAL CALCULUS 

13. / (x) = x^i^ ^. f (x) = a^sin X (sin x/x + log a; • cos x). 

14. / (x) - (sin a;)^. /' (x) = (sin x)^ {x cot x + log sin x). 
16. / (0) = (sin (?)tan e. /' (0) = (sin 0)tan (1 4-sec2 6 log sin 0)., 
16. /(0) = itan3 0-tan0+0. /' {$) = tan^^. 

-(sec Vl - xy. 



17. / (x) = tan VI - a;. /' (x) = ^ 

2 Vl -X 



\/f^ 



18. / (0) = log \/ : . -- - f (6) = CSC d. 

By differentiation derive each of the following pairs of identities 
from the other: 

19. sin 2 s 2 sin d cos d, cos 2 = cos^ d — sin^ 6. 

„^ . ^ ^ 2 tan ^ ^ 1 - tan2 q 

20. sm 2d = - — -- — — : , cos 2 



l+tan2 0' l+tan2 

21. sin 3 = 3 sin - 4 sin^ d, 
cos 3 = 4 cos3 - 3 cos 0. 

22. sin (w + n) = sin md cos nd + cos mO sin ri^, 
cos (m -\- n) 6 = cos m0 cos n9 — sin md sin n-0. 

23. If 6 vary uniformly, so that 360° is described in tt seconds, show 
that the rates of increase of sin 6, when 6 = 0°, 30°, 45°, 60°, 90°, are 
respectively, 2, V3, V5, 1, 0, per second. (See figure. Art. 52.) 

53. The Sine Curve or Wave Curve. — The locus of the 
equation 

y = smx, (1) 

where x is an angle in radians, is called the sine curve, from its 
equation, or the wave curve, from its shape. The maximum 
value of y is called the amplitude, being unity in (1); and, 
since the curve is unchanged when x + 2 tt is substituted for 
X, the curve y = sin x is a periodic curve with a period equal 
to 2 TT. (See figure, Art. 52, and figure, Art. 73.) 
The more general form of the equation is 

2/ = a sin mx, (2) 

where a is the amplitude and — is the period, m a constant. 

The curve is called the sinusoid also, and is of great 
importance, since it is the type form of the fundamental 
waves of science; such as, sound waves, vibrations of rods, 



DAMPED VIBRATIONS 73 

wires, plates and bridge members, tidal waves in the ocean, 
and ripples on a water surface. The ordinary progressive 
waves of the sea are not of this shape, as they have the 
form of a trochoid. 

54. Damped Vibrations. — When a body vibrates in a 
medium Uke a gas or hquid, the amphtude of the swings get 
smaller and smaller, or the motion slowly (or rapidly in some 
cases) dies out. Thus, when a pendulum vibrates in the air 
the rate of decay of the amplitude is quite slow; but when 
in oil the rate is rapid. The ratio between the lengths of the 
successive amplitudes of vibration is called the damping 
factor or the modulus of decay. 

In all such cases the amplitude of the swings differ hy a 
constant amount or the logarithmic decrement is constant. 
Hence the amplitude must satisfy an equation of the form 

A = ae-^', (1) 

where A is the amphtude and t the time. The actual 
motion is given by an equation of the form 

V 

y = ae~^^ sin oit, where co = - is a constant. (2) 
■ a 

(See Art. 73.) 

* In plotting a curve whose equation is of this form, say, 

2/ = e-^^ sin I a;, (3) 

much is gained by the following considerations: 

1. Since the numerical value of the sine never exceeds- 
unity the values of y in (3) will not exceed in numerical value 
the value of the first factor e~*^. As the extreme values of 
sin^Tra; are +1 and —l,y has the extreme values e~*^ and 
— e~^^. Hence, if the curves 

2/ = e-*^ and y = — e"^^ (4) 

* This illustration is given substantially in Smith and Gales's New 
Analytic Geometry. 



74 DIFFERENTIAL CALCULUS 

are drawn, the locus of (3) will lie entirely between these 
curves. They are called boundary curves, and they are 
plotted by three or more points, the second being symmetri- 
cal to the first with respect to the x-axis. 




2. When sin ^ttx = 0, then in (3) y = 0, since the first 
factor is always finite. Hence, the locus of (3) meets the 
X-axis in the same points as the sine curve 

2/ = sin J TTX. (5) 

3. The required curve is tangent to the boundary curves 
when the second factor, sin J ttx, is + 1 or — 1 ; that is, when 
the ordinates of the curve (5) have a maximum or a minimum 
value. The tangency is proven by finding the derivative of 
y in (3) and noting that, when sin | ttx is +1 or — 1, it will be 
the same as the derivatives of y in (4). Hence, the slopes of 
the curves and the ordinates being equal for the same values 
of X, the required curve is tangent to one or the other of the 
boundary curves for those values of x that make sin J ttx = 
+ 1 or —1. Thus, differentiating (3) and (4) gives 

dy 1 , . 1 , TT ,^ 1 

-J- = — -r e~^ ^ sm -TX -\- -x e~i ^ cos ^z tx 
dx 4 2 2 2 

= — ie-^"", when sin J ttx = 1, 

= i e~^ ^, when sin ^ ttx = — 1 . 

For the sine curve (5) the period is 4 and the amphtude is 1. 
This curve is the broken line of the figure. 

The locus of (3) crosses the x-axis at x = 0, d=2, ±4, ±6, 



DAMPED VIBRATIONS 75 

etc., and is tangent to the boundary curves (4) at x = ±1, d=3, 
db5, etc. The discussion having disclosed these facts, the 
curve is readily sketched, as in the figure; that is, the wind- 
ing curve between the boundary curves (4). 

A more general form of the equation of a damped vibration 
is 

y = ae~^^ sin (cat — a), where a = — ^ is constant. (3') 

a 

This equation may be written either (see Art. 73) 
y = e~^^ (A sin kt -\- B cos kt), 
where A and B are constants, 
or y = A sin {cct — a), where A = ae-^K (3'0 

Here A is a variable decreasing amplitude, whose relative 
rate of decrease is —dA/dx -^ A = b; that is, the relative 
rate of decrease of A is constant. 

The successive derivatives from (30 are (by Art. 68): 

dy 

-jT = ae~^^ [— b sin {oit — a) + co cos {wt — a)], 

—^ = ae~^^ [¥o)'^ sin (oit — a) — 2bo) cos {o)t — a)], 

whence it follows that 

g + 26|+(6^ + co^), = 0. (40 

Equations which contain derivatives or differentials are 
called differential equations. The equation (4^ is the funda- 
mental differential equation for damped vibrations. The 

dy 
term in -7-, or v, proportional to the velocity, occurs in equa- 
tions for vibration only when damping is considered. Vibra- 
tions are cases of simple harmonic motion — damping being 
caused by resistances, such as friction, etc. Simple har- 
monic motion is treated in Art. 73. 



76 DIFFERENTIAL CALCULUS 

INVERSE TRIGONOMETRIC FUNCTIONS. 

55. Formulas and Rules for Differentiation. — The 

direct trigonometric functions are single-valued but the 
angle has to be restricted to a certain range in order that 
the inverse functions may be single valued. To make 
the inverse functions single- valued, the angle denoted by 
sin~i ^^ cosec"^ x, tan"^ x, cot~^ x, covers"^ x, is taken to lie 

between — - and ^ , and the angle denoted by cos~^ x, sec~i x, 
vers~^ X, to lie between and t. Thus 



i^l) = ^^cos-i(^); sin-i(-^)=-|; 



sm-^ I - 1 = - = cos-^ V~2~/ ' ^^^~^ ( ~" 2/ 

cos M ^ i ^ "fi" ' sm 1 X + cos ^ x = ^ ; 

tan~^ X + cot~^ ^ = 9 , if a; be positive, 

= — ^ , if a; be negative. 

These restrictions will be assumed in the following formulas, 
and all will be expressed in terms of the letter x. While the 
symbols sin~^ x and arc sin x are both used to denote the 
angle whose sine is x, in writing the formulas the notation 
sin"i X is preferable. 

[XVII] d sin-^ :i; = -7^=. 

The differential of an angle in terms of its sine is the differen- 
tial of the sine divided by the square root of one minus, the square 
of the sine. 



[XVIII] dcos-^ 



dx 

X = — 



Vi- 



The differential of an angle in terms of its cosine is minus 
the differential of the angle in terms of its sine. 

[XIX] dtan-'* = j^. 



DERIVATION OF [XVII] AND [XVIIIl 77 

The differential of an angle in terms of its tangent is the 
differential of the tangent divided hy one plus the square of the 
tangent. 

rlv 

[XX] dcor'..= -5^. 

The differential of an angle in terms of its cotangent is minus 
the differential of the angle in terms of its tangent. 

fix 

f XXI] d sec-^ :*: = , , . 

The differential of an angle in terms of its secant is the differ- 
ential of the secant divided by the secant and the square root of 
the square of the secant minus one. 

fix 

[XXII] d cosec-^ x= " . 

^ xVx^-1 

The differential of an angle in terms of its cosecant is minus 
the differential of the angle in terms of its secant. 

fix 

[XXIII] d vers-^ :*: = , ■ 

The differential of an angle in terms of its versine is the 
differential of the versine divided by the square root of twice the 
versine minus the square of the versine. 

dx 



[XXIV] -, / 

The differential of an angle in terms of its cover sine is minus 
the differential of the angle in terms of its versine. 

66. Derivation of [XVII] and [XVIU]. — 

Let d = sin~i x; then sin $ = x, the differential of which 
by [XI] is cos 6 dd = dx; 

dx dx dx 



cos (9 Vl-sin^^ Vr 



Now d cos~i X = dl^— sin~i xj= 7= 



78 DIFFERENTIAL CALCULUS 

57. Derivation of [XIX] and [XX]. — 

Let 6 = tan~i x; then tan d = Xf the differential of which 
by [XIII] is secHdd = dx; 

iQ _ dx _ dx _ dx 
•*• ~ sec2^ ~ l+tan2 6> ~ 1 + x^' 

Now d cot~^ X = di^ — tan~i x] = — r— ; — 5- 

\2 / 1 + x^ 

58. Derivation of [XXI] and [XXII]. — 

Let 6 = sec~^ x; then sec 6 = ic^ the differential of which 
by [XV] is sec 6 tan 9dd =dx; 

dx dx 



dd = 



sec 6 tan d x v sec^ d — 1 
dx 



^ Vx2 - 1 

Now d cosec~i x = dl^ — sec"^ x] = , 

V2 J X Vx^ - 1 

59. Derivation of [XXIII] and [XXIV]. — 

Let 6 = vers~^ x; then vers = x, the differential of which 

by Cor., Art. 45, is sin d^ = dx; 

,- rfx c?x dx 

do = 



sin Vl - cos2 Vl - (1 - vers oy 
dx dx 



Vl-{l-xy V2x-x^ 

Now (i covers"^ x = dl-p: — vers"^ a; )= , 

\2 / V2a;-x2 



1. d sin-i 



EXERCISE VI. 

X _ d (x/a) _ dx 
« " Vl - (x/a)^ ~ V'aF^^' 



- , .a; —dx J, ,x adx 

2. d cos * - = , : d tan^ 



J ^ .X —adx J _,x adx 

d cot~i - = -IT-. — s ; d sec ^ - = 



a a^ + x^' ax Vx"^ - a^ 

, , a; —adx -, .x dx 

d csc~^ - = . ; d vers ^ - = 



Vx"^ - a2 ' « V2 aaj - 

iVoie. — These may be considered standard formulas. 



EXERCISE VI 79 

« . i , dy ^ . , , tanx 

3. 2/ = tan x tan~i ^^ ^ = gecs x tan"i x + — - — :• 

A -f -1 2a; dy ^ 2 (1 - rc^) 

2/ tan j,_j_^2- ^^ l+6a;2+a;4* 

c • -1 2; + 1 c?2/ 1 

V 2 da; Vl - 2 X - a;2 

« . ^/- • dy Vl+csca;. 

6. 2/ = arcsm Vsina;. 3^ = 

^ da; 2 



7. y =x^ 



S. y = tan~i (n tan a;). 



9. 2/ = arc cos 



e-^ — e' 



— X 



10. y = arc vers 



2a;2 



1 +a;' 



11. 2/ = arc tan :j ^-r," 

1 — o X' 

1 — X^ 

12. 2/ = arcsinj--jj^- 

10 , a; + a 

13. 2/ = arc tan 



1 — ax 



dy 
14. (f> = arc tan -r- • 



15. 2/ = tan ^ ■ 

VI - 



16. 2/ = arc sin 



^ = ^sin-i X / sin ^x _^ logx Y 
da; V ^ Vl-xy 

dy __ 



17. 2/ = arc tan (sec x + tan a;). 

^ o • /sin a; — cos a;\ 

18. 2/ = arc sm f 7= • 

V ^2 y 

ia ^ _i3a;-2 , ^,3a;-12 d2/ a 

19. 2,=tan^^-+cot^g^^^. ^ = 0. 

20. 2/ = arccot^-5,-:3^,. ^ = ^-,,^^:^ 



dx' 


C0s2 X -\-n sin2 x 


dy 


-2 


dx 


e^ + e-^ 


dy _ 

dx~ 


2 

~ l+x^' 


dy 


3 


dx 


l+a;2 


dy 


-2 


dx 


1+^2 


dy 


1 


dx 


l+a;2 




d^y 


d<t> 


dx^ 


dx 


■*©■ 


dy 


1 


dx 


VI - X2 


dy 


-2 


dx 


e^ + e-* 


dy 
dx 


1 
2' 


dy 
dx 


= 1. 



80 



DIFFERENTIAL CALCULUS 



21. What is the slope of the curve y = sin x? Its inclination lies 
between what values? What is its inclination at a; = 0? What at 
x=x/2? 

The slope = cos a:; hence, at any point, it must have a value between 
— 1 and +1, inclusive. Hence, the inclination of the curve at any point 
is between and 7r/4 or between 3 7r/4 and tt, inclusive. (See figure, 
Art. 52.) 

^ 60. Hyperbolic Functions. — These are certain functions, 
recognized as far back as 1757, that have been introduced in 
recent years, and that are coming more and more into use. 

As the trigonometric functions are called circular because 
of their relation to the circle, the hyperbolic are so called 
because of their relation to the rectangular hyperbola, the 
relations being in some respects the same. The functions 
are analogous to the trigonometric functions and their names 
are the same. They are the hyperbolic sine, cosine, tangent, 
etc., and they are defined as follows: 



sinhic 



\{e'-e- 



cosh X = ^ (e^ + e~^ 



tanh X = 



e^ — e~ 



cscha; 



sech X 



coth X = 



sinha; 
1 

cosh a; 

6^ + e-' 



61. General Relations. — Besides the reciprocal rela- 
tions given above, the same as those between the circular 
functions, there are analogous relations : 

cosh^o; — sinh^a; = 1; 1 — tanh^x = sech^x; 

coth^ X — 1 = csch^ a;; sinh 2 a; = 2 sinh x cosh x; 

cosh 2 x = cosh^ x + sinh^ x = 2 cosh^ —1 = 1+2 sinh^ x. 

62. Numerical Values. Graphs. — The sine may have 
any value from — oo to oo ; the cosine any value from 1 to 

y. 
I 




7/=tanhJC 



. INVERSE FUNCTIONS 81 

00 ; the tangent any value between —1 and 1, and the Hnes 
whose equations are y = =tl are asymptotes to the graph of 
tanh X. The graphs of the sine and cosine are both asymp- 
totic to the graph oi y = ^ e^. 

63. Derivatives. — Since j- e* = e* and j- e"^ = — e"^, 

ax dx 

by differentiating the hyperboUc functions as functions of ic, 
the several derivatives are readily found to be as follows: 

-T-sinhx = coshx; -^-coshx = sinha;; 

-T-tanhx = sech^x; -^-cothx = — cosech^x; 

-J- cosech X = ~ cosech x coth x ; 

-T- sech X = — sech x tanh x. 
ax 

The differentials are given at once by the derivatives, or 

vice versa; thus, d sinh x = cosh x dx, and so for the others. 

64. The Catenary. — The curve y = cosh x = J (e^4-6"^) 
is called the catenary and is important because it is the 
curve of a perfectly flexible and inextensible cord between 
two points, and is the curve that a material cable when hung 
between two supports is assumed to take. 

Since -r- cosh x = sinh x, l^ = o (^"^ ~ ^~'') = sinh x 

is the slope of the catenary. The general equation of the 

catenary is 

. X af ^^ -l) 
y = a cosh - = o U" + 6 "/, 

where a is the distance from the origin to the lowest point 
of the curve. (See Art. 146.) 

65. Inverse Functions. — The inverse functions are 
useful when expressed as logarithms. 

li y = sinh~^ x, the logarithmic form of y is found from 
X = sinh y = i(ey — e~y), 



82 DIFFERENTIAL CALCULUS 

which is reduced to 

e^y -2xey - 1 = 0; 

solving as a quadratic gives 

ey = x± VxM-T; 
but as ey is always positive, 

gj/ = X + Vx^ + l ; .-. sinh-i x = y^\og{x-{- Voj^+l). 
In the same way is found, cosh~^ x = log (x ± Vx^ — 1 ) • 

Since (x - ^W^l) = (^qrvP^T) ' 

/. log {x - ^x^ - l) = -log (x + Vx2- l). 

For each value of x greater than 1 there are two values of 
cosh~i X, equal numerically but of opposite sign. 
In the same way again is found, 

tanh-i X = - log , If x^ <\\ 

2 \ — X 

COth~l X = p: log -, if x^ > 1. 

2 ^x — 1 

66. Derivatives of Inverse Functions. — The derivatives 
of the inverse functions are found by differentiating their 

logarithmic forms, using formula -r-logx = -• 

O/X X 

X . 

The derivatives, taking - instead of x, are: 

d . , . X 1 

sinh~^ - = 



dx a Vx^ + a^' 

-1- cosh~i - = ± 



dx a Vx2 - a2 ' 

-J- tanh-i - = -T 5, (x^ < a^) ; 

dx a a^ — x^ ^ 

|-coth-i- = -f^, (a;2>a2). 
dx a x^ — a^ 



DERIVATIVES OF INVERSE FUNCTIONS 83 



Since the inverse cosine is not single- valued, for the positive 
ordinate of cosh~^ 
is used instead of x, 



x X 

ordinate of cosh"^ -, the + sign must be taken. When - 



. , 1 X 1 x-\- Vx^ + a^ 

sinh~^ - = log 

a * a 

= log {x + VxM-~^) — log a, 

X 

so the derivative of sinh~i _ jg ^j^g same as that of log 
a 

{x + Va;2 -\- o?), since d (log a) = 0. The divisor a occurs in 

the logarithmic form of cosh"^ - also, so its presence should 

be borne in mind when comparing the same result expressed 
in logarithms and in inverse hyperbolic sines or cosines. 

The relation of the inverse hyperbohc sine to the equi- 
lateral hyperbola is shown in Art. 137, and the inverse func- 
tions are again considered in Art. 120. 



CHAPTER III. 

SUCCESSIVE DIFFERENTIATION. ACCELERATION. 
CURVILINEAR MOTION. 

67. Successive Differentials. — It is often desired to 
differentiate the differential of a variable or to get the deriv- 
ative of a derivative. For, while the differential of the in- 
dep)endent variable, being arbitrary, is usually supposed to 
have the same value at all values of the variable and hence to 
be a constant, the differential of the dependent variable, 
except when the function is Hnear, is a variable, subject to 
differentiation. 

The differential of dy is called the second differential of y; 
the differential of the second differential of y is called the 
third differential of y; and so on. d{dy) is written d^y; 
d{d^) or dd dy, is written d^y; and so on. The figure written 
like an exponent to d denotes how many times in succession 
the operation of differentiation has been performed, dy, 
d^y, d^y, , . . d^'y are called the successive differentials of y. 
Example. — The successive differentials of y when y = ax^: 

dy = dax^dx; 

d^y = Zadx ' d (a;^) = 6ax dx^; 

d^y = 6 a dx^ * dx = Qa dx^; 

d^y = d(Qa dx^) == 0. 
The independent variable being x, dx is treated as a constant. 
Note that according to the notation adopted d^y = ddy; 
dy'^^idyY; d(y'^)^2ydy. 

68. Successive Derivatives. — The derivative of the 
first derivative of a function is called the second derivative of 
the function; the derivative of the second derivative is 
called the third derivative; and so on. 

84 



SUCCESSIVE DERIVATIVES 85 

When X is independent, 
d^dy^d^y d_d?y ^^y^ d^ d^'-^y ^ d^'y 

dx dx ~~ dx^' dx dx^ ~ dx^' • • * ' ^^ dx""'^ ~ dx"' 

The successive derivatives of / (x) are denoted by 

fix), r(x), rix), r{x), . . . ,f^{x). 

Thus if / {x) = x\ f (x) = 4 x^ }" {x) = 12 x", 
S"'{x) = 24.x, f^ (x) = 24, r (x) = 0. 

Hence, ii y = f (x) and x is independent, 

dx ^ ^''^' dx'' ^ ^""^^ ' ' ' ' dx- ^ ^'^^• 

The nth derivative of some functions can be easily found by 
inspection of a few of the derivatives. 

Example l.—f{x) = e'^J'ix) = e'^J^ix) = e^,J"'{x) = e^, 
. . . , .-. /" ix) = e\ 
This function e^ is remarkable in that its rate of change, or 
derivative, is equal to the function itself. 

Example 2.—f{d) = sin 61, f (d) = cos ^, f (d) = -sin 6, 

f" (0) = -cos 6, /^^ (d) = sin (9. /' (6) = cos6' = sin (d + ^), 

r (d) = COS {^ + l) = sin (^ + 2 • ^) , r ie) = cos (d + ir) 

= sin(^ + 3.|) . . . ; .•./"W=sin(0 + n.|). 

Each of the successive derivatives of f (x) equals the x-rate of 
the preceding derivative, for f"" (x) = -j- /"~^ (x) = the a:-rate 

of/-i(x). 

Corollary, — /""^ (x) is an increasing or a decreasing 
function of x according as /" (x) is positive or negative, and 
conversely. 

Note. — The tangential acceleration is, 
_ dv _ d ds _ d^s 
^'~Jt~dtdt~d^' 



86 DIFFERENTIAL CALCULUS 

and the flexion is 

, _ dm _ d dy _ d^y 

dx dx dx dx^ 

ds 
(See Art. 12 and Art. 13.) Hence, the speed -^ is increasing 

dh 
or decreasing according as the acceleration jr^ is positive or 

negative, and the slope is increasing or decreasing according 

d^u 
as the flexion j^ is plus or minus. 

When the second derivatives are equal to zero, the first 
derivatives are constant, or conversely. (See Art. 13.) 

69. Resolution of Acceleration. — An acceleration, like 
a velocity, being a quantity which has magnitude and 
direction, may be represented by a straight line, that is, by 
a vector. 

In general the acceleration a at any point (x, y) of a curvi- 
linear path may be resolved into two 
components in given directions. The 
directions usually taken are along the 
tangent and normal at the point, and 
in directions parallel to rectangular 
axes OX, OY. With the notation of 
the figm-e for (d). Art. 10, the com- 
ponents parallel to the axes being the 
rates of change of dx/dt and dy/dt will be denoted by d^x/dt^ 
and dhj/df^, respectively. The rate of change of the velocity 
is the resultant acceleration 




-^7TV = ^(f)V(tJ 



To find the component acceleration at along the tangent 
at P; resolve the axial accelerations along the tangent, giving 
for the sum of tangential components, 

d^x , d^y . , dH dx , d^y dy d^s 



EXERCISE VII 87 

by differentiating, 



\dtj \dt) '^{dtj 



^t= -jT^ = -iT = rate of change of the speed. 

Hence, the tangential component is the same as for recti- 
hnear motion. 

EXERCISE VII. 

Find dy, d^y, d^y, when: 

1. y = 2 3^ - 5 x' + 20 x^ - 5 x^ + 2x. 
d?y = 120 (x2 - x + 1) dxK 

o 9 1/ IN J. 2(x^ -Sx + 3) , , 

2. y = x^ log (x - 1). d^y = ^— tt^—^ dxK 

{X — ly 

3. ?/ = (x2 - 6 X + 12) e^. d^y = x^e^ dx\ 

4. ?/ = log sin x. d^y = 2 cos a; sin-^ x dx^. 

5. 2/ = tan X. d^y = (6 sec^ x — 4 sec^ x) dx^. 
Find the successive derivatives: 

6. fix) =a^ + 4.x' + Sx + 2. /vi(x)=[6, /vii(^)=o. 

7. / (x) = log (1 + x); find nth derivative. 

/' (x) = (1+ a;)-S /" (a:) = (-1) (1 + x)-^ 
r"{x) = (-1)2L2_(1 +x)-3, P^{x) = (-iy\s_{i+x)-s . . . 
.-. r (x) - (-1)"-! |n-l (1 + a:)-^ 

8. fix) = a^Uoga;. /^^ ix) = 6x-K 

d^y _ Q e^ — e~^ 



9. 2/ = log(e^ + e-^). 



dx=« (e^ + e-^)3 

10. Find formula, known as Leibnitz's theorem, for d^ (uv). 
Let u and v be functions of x; then 

d (liv) = du'V-\-udv, (1) 

^2 (wy) = d^u 'V-\-dudv + dudv-\-u d^v 

= d^U'V + 2dudv + ud^v ; (2) 

.-. d^ iuv) =d^U'V + 3d^udv + d du d^v + u d^v. (3) 

The coefficients and exponents of differentiation are according to the 
Binomial theorem, however far the differentiation is continued; 

.-. d" iuv) = d'^w . y + nd^-i udv -]- ^ ^^ ~ ^^ ^^^-2^ d^y + • • • 

+ wdw d'*"^ y + w d" y. 



88 



DIFFERENTIAL CALCULUS 



11. If a;2 + ?/2 = a^, 



^3-If^. + |-. = l^ 



d2y 
dx2 



12. If ,^ = 2 p., g= ^, 






14. 



a2 62 



^ 
dx2 



1 !f_»_ __r_ 



3 
64_ 



70. Circular Motion. — When a point describes a circle 
of radius r with constant speed v^ it has a constant accelera- 
tion v^jr directed towards the center of the circle. 

Let FT be the velocity at P, 
and PiTi that at Pi. A velocity 
being a directed quantity may be 
represented by a vector; that is, 
by a straight line whose length 
denotes magnitude and whose 
direction is the given direction. 
Hence from a common origin o, 
the vectors op and opi are drawn 
equal to the vectors FT and 
PiTi, respectively. Since the 
speed is constant each vector is 
increment, denoted by tS.v. The 

Ay 




v^ and ppi is the vector 
average acceleration for 



the interval of time A^ 



A^ 



directed along ppi, and is laid off as pm. 

As Ai approaches zero. Pi approaches P, and pi approaches 
p along the circular arc indicated by the dotted line; 'pm 
approaches a vector p^ directed along the tangent to the arc 

ppi at p. This vector, the lim -r-r L represents the accelera- 

tion -TT of the point P moving in the circle of radius r; and 

since the direction is at right angles with the tangent at P, 
the acceleration is directed towards the center 0, is normal 
acceleration, therefore, denoted by «„. To find the magni- 
tude of the normal acceleration a^. since the sectors popi 
and POPi are similar, the angles at o and being equal, 



CIRCULAR MOTION 89 

arc ppi _ arc PPi arc ppi _ As . 

op OP ' V r 

arcppi V As 



, and limr^^V^limr^li 



At r At 

replacing the arc ppi by its chord, (Art. 22.) 

,. rchordppil _ dv _vds 
A^<2i L ^i j~ dt ~ r dt' 

dv v^ ,^. 

•■ «" = d« = r (1) 

Otherwise, it may be seen that while the point P describes 
the circle of radius r, the point p describes the circle of radius 
V, the velocity of p in its path being the acceleration of P in 
its path. Since the circles are described in the same time, 
the velocities are to each other as the paths of the two points, 
or as the radii of the circles. 

velocity oi p _ velocity of P . 

V r 

dv 
velocity of p is -r:, rate of change of velocity v of P, 



an V 


v^ 




an = - 


V r 


r 



Since the speed is constant, the rate of change at is zero, 
.*. a = Vat^ + a„2 = a„ = - ; that is, the total acceleration 

is the normal acceleration, the change of velocity being change 
of direction only. Since 

s = 27rr = vt; a = -^, (2) 

where T is time of a revolution. 

Note. — By Newton's Second Law the measure of the 

W 
force on a moving body is — a (Art. 71); hence, the force 

acting on a body weighing W lbs. revolving in a circle of 



90 DIFFERENTIAL CALCULUS 

radius r is pounds of force, is directed towards the 

gr 

center of the circle, and is called centripetal force. The 

reaction of the body to this force is by the Third Law equal 

in magnitude and opposite in direction. It acts upon the 

axis or upon whatever deflects the body from its otherwise 

rectilinear path, and has been called the centrifugal force, 

although a misnomer. The centripetal force is the active 

force, the other is the equal and opposite reaction and should 

be called the centrifugal reaction, since it -is the resistance 

which the inertia of the body opposes to the force acting 

upon it. 

71. The Second Law of Motion. — According to New- 
ton's Second Law of Motion the rate of change of momentum 
of a body is propiortional to the resultant of the impressed 
forces acting on the body. 

Let a body of standard weight W be moving with velocity 
V, then Wv = momentum of the body; 

^^KVVV) VY ^^ VY ^^^ YYat. 

Hence, if F be the resultant force and a the acceleration, by 
the Law, 

WaocF or Wa = kF', (1) 

that is, the product of the numbers representing the weight 
and the acceleration is proportional to the number represent- 
ing the force. The value of the factor k depends upon the 
units used for the other factors. When these are the usual 
units, foot, pound (weight), second, pound (force), it is 
found by experiment that h has the value 32, approximately. 
Experiment shows that, while the value varies slightly for 
different localities, it is the same for all bodies in any one 
locality. This value is denoted by g and is called the accel- 
eration of gravity*; for when a body falls freely, gravity being 
the only force acting, the acceleration is found to be about 



THE SECOND LAW OF MOTION 91 

32 feet per sec. per sec. The locality in which g = go = 
32.1740 ft./sec.2 has been adopted as the ''standard locality" 
and the weight of the body in that locality is called the 
standard weight of the body. Putting g for k in equation (1) 
it becomes 

W 

Wa = gF or F = ~a. (2) 

g 

If the force F is the force of gravity acting on W, then 

dv 

* = « = »' 

the acceleration of gravity; for the weight W is the force of 
gravity acting on the body denoted by the letter W. 

Since for any given body the ratio of the force to the 
acceleration produced is constant, the value of this ratio, 
F/a or W/g, is a characteristic of the body, called its inertia 
and the ratio may be denoted by the letter m; then the 
equation (2) may be written 

F = ma. (3) 

In using equation (3) for the solution of problems, with the 
usual units, m must be replaced by W/g. 

Some writers use the word ''mass" to denote the inertia, 
while others use it for standard weight; consequently, there 
are some who avoid the use of the word on account of the 
resulting confusion. 

In Physics the equation (3) is used, the unit of force, called 
the absolute unit, being that unit which in equation (1) 
makes k = 1, the other units remaining the same, and, 
therefore, m measured in pounds the same numerically as 
the standard weight W. 

Accordingly, if the nimiber g be the number of absolute 
units of force with which gravity attracts the unit mass (or 
weight), the Law becomes 

m -^ = mg, hence -tl = g, the acceleration of gravity. 



92 DIFFERENTIAL CALCULUS 

The absolute unit of force is thus, that force, which acting on 
the unit of mass (or weight) for the unit of time, generates the 
unit of velocity. The absolute unit of force is thus 1/g of a 
pound avoirdupois, about J of an ounce, and F is given in 
this unit when in 

F = ma, 

m is expressed in pounds, the unit being a pound. The 
ordinary unit of force, sometimes called the Engineer's unit, 
is one pound and is g times the absolute unit used in Physics. 
Newton's Second Law of Motion gives as a definition of 
force: force is the time-rate of change of momentum. Using 
the much abused term ''mass," the definition is: the force 
is the product of the mass times the acceleration. From 

W 
F = ^a', (2) 

y 

J, _W d's ' _ Wv' 

g dt^' gr 

^WdH ^Wd^y 

^ g dt^' ' g dt''' 

the tangential, normal, and axial components of a force F, 
corresponding to the accelerations, at, an, ax, ay. Since 
kinetic energy of a moving body is E = J mv'^, 

dE 

-r- = mv, 
dv 

that is, the i^-rate of E is momentum; 
dE dv j^ 

that is, the time-rate of E is product of force and velocity. 

72. Angular Velocity and Acceleration. — When a body 
is rotating about an axis the amount of rotation depends 
upon the time; so if B is the angle through which any line in 
the body, intersecting the axis at right angles, turns, then d 



ANGULAR VELOCITY AND ACCELERATION 93 

gives the amount of rotation and is a function of the time t. 
Thus in the case of a wheel the rotation is measured by the 
angle d through which a spoke turns in a time t. The rota- 
tion is imiform if the body rotates through equal angles in 
equal intervals of time. The rate of rotation. or the rate of 
change of the angle is the angular velocity or speed and is 
denoted by co. 

If the rotation is uniform, the angular velocity is constant 

and 0) = -, 6 being in radians; hence, if the uniform rate of 

rotation is co radians per second, the body rotates through cot 
radians in t seconds of time. 

If the rotation is not uniform the rate at which the body 
is rotating at any instant is the angular velocity at that time, 

, ,. A^ dd 

and CO = lim tt = -r; • 

Af=0 ^t dt 

This expression for angular velocity is general and is applic- 
able when the rotation is uniform also; for then, 

^e ^M ^dd 
"^ ~ t~ M~ dV 

although, the ratios being constant, no hmit is involved. 

Similarly, the angular acceleration is a = -, for constant 
acceleration ; and 

'^~ dt~ dt [dt ~ df" 



is, in general, the time-rate of change of angular velocity, 
or the angular acceleration. 

If a particle is at a distance r from the axis of rotation, the 
relation between the angular velocity of the particle and its 
linear velocity follows at once, whether the rotation is uni- 
form or not. 



94 



DIFFERENTIAL CALCULUS 



Since in circle 


As, = r . AS, 




A6> 1 As 




M~ r ' At' 




,. A^ 1 ,. As 
lim -irr = - ' lim tt 
A<=o At r At=o At 


••• 


_dd _1 ^ds _v 
dt r dt r 




the relation sought. Hence, since 
the angular velocity of every point 
of a rotating body has the same 
value at any instant, and the direc- 
tion of motion of a particle at any 
point is along the tangent at the 
point, 

tangential velocity v = rco, 
if CO is the angular velocity of the 
particle about axis at 0. 



Since 



dv 
dt 



d{ro)) 
dt 



or 



d^ ^ d^ 
dt^ ~^dt^' 



ra. 



.'. tangential acceleration at 

the relation between tangential acceleration and angular 
acceleration when a is the angular acceleration of a particle 
at a distance r from the axis of rotation. 



Since 



dv v^ r^bP- „ 

ttn = -77 = — = = rC0% 

dt r r 
normal acceleration an = ro)'^. 



73. Simple Harmonic Motion. — If a point move uni- 
formly on a circle and the point be projected on any straight 
Hne in the plane of the circle, the back-and-forth motion of 
the projected point on the given straight line is called simple 
harmonic motion. It is denoted by the letters S. H. M. 
Let the point P move upon the circumference of a circle of 
radius a with the uniform velocity of v feet per second, so 



SIMPLE Hi^RMONIC MOTION 



95 



that the radius OP rotates with uniform angular velocity at 
the rate of - = co radians per second. The projection, P\ 



a 



of P on the vertical diameter, moves 
up and down. Let d be the angle that 
the radius makes with the o^-axis, then 
if the point P was at A when t = 0, the 
displacement OP' = y is given by 

?/ = a sin = a sin oit. 

If the point P was at Po when t = 0, and* at A when t = U 




then 



y = a sin {ojt — a) , where a = coto = 



vto 



(1) 



When the displacement at time t is given by (1) the motion 
is S. H. M. Hence, the point P' describes S. H. M. 

The velocity of a point describing S. H. M. is, from (1), 
dy 



dt 
and the acceleration is 



= oco cos (ojt — a). 



(2) 



d^v 

^^^ = aa;2 sm (co« a) 


(3) 


= -oi'^y, from (1), 


(4) 

(5) 



or 



It should be noted that equation (1) may be written in the 
form 

y = a sin (cot — a) = a [sin o)t cos a — cos oit sin a] 
or y = A sin kt -\r B cos kt, 

where A = a cos a and B = —a sin a are constants. This 
equation and y = asin (kt — a) are the general formulas 
for S. H. M. 

The acceleration of a particle describing S. H. M., as 
shown by (4), is proportional to the displacement and 



96 



DIFFERENTIAL CALCULUS 



oppositely directed. It is oppositely directed since the 
motion is one of oscillation about a position of equilibrium. 
When the body is above this position the force is directed 
downward, and when it is below, the force is upward. In the 
figure the point P' has a negative acceleration when above 
and a positive acceleration when below 0. The acceleration 
is zero at 0, a maximum at B' and a minimum at B) while 
the corresponding velocity, as given by (2), has its maximum 
numerical value as P' passes through in either direction, 
and is zero at B and B' , the ends of the vertical diameter. 
The factor of proportionality co^ is connected with the 'period 

2'jf 
T by the relation T = — , where the period of the S. H. M. 

y = asm o)t is the time T required for a complete revolution 



.of the point P; that is, coT = 2 7r. The time i 



- - to 

CO 



make part of a revolution is called the phase, a being epoch 
angle. The number of complete periods per unit of time 

is iV = 7p = 5—, where N is the frequency of t'he S. H. M. 

Let P' be a tracing point capable of describing a curve on a 
uniformly translated sheet of paper, SS', then if the sheet be 



>\— 7^' 


-ry^-/^ 


M] fe,' 


-A-x 


\ 




'^f-M 


pV p\o/ 


•^Kk- 


^-P'j^ 



moved with the same speed as the point P moves on the 
circumference of the circle of radius a, P' describing S. H. M. 
on the vertical diameter will trace the sinusoid P'BP'B' on 
the moving paper. The sinusoid will have as its equation 



y 



a sm - 1 
a 



asm 



where x is the abscissa of any point of the sinusoid referred 



EXERCISE VIII 97 

to an origin (as 0') moving with the paper. The circle is 
shown in the figure in several positions corresponding to the 
different angles through which the radius OP has revolved, 
or the different positions of the projected point P' on the 
vertical diameter BOB\ The amphtude of the S. H. M. is 
the same as that of the sinusoid; that is, the radius a of the 
circle. The period of the sinusoid is 2 ira, correspondii^ to 

the period, T = — , of the S. H. M. of the point P' on the 

vertical diameter. 

74. Self -registering Tide Gauge. — The principle by 
which the up-and-down motion of a point is represented by 
a curve is utiUzed in the self -registering tide gauge for record- 
ing the rise and fall of the tide. Such a gauge consists 
essentially of a float protected by a surrounding house or 
tube, and attached by suitable mechanism to a pencil that 
has a motion proportional to the vertical rise and fall of the 
float. The pencil bears against a piece of graduated paper 
fastened to a drum that is revolved by clockwork. There 
will thus be drawn on the paper a curve where the horizontal 
units are time, and the vertical units are feet of rise and fall. 
The stage of the tide is given for any time. 

EXERCISE Vin. 

1. The angle (in radians) through which a rotating body turns, 
starting from rest, is given by the equation 

= 1 ai2 _j_ ^^^ _|_ 0^^ 

where a, wo, 6o are constants; find the formulas for angular velocity and 
angular acceleration after any time t. 

CO = -37 = at -\- m, which gives the angular velocity; 

-^ = ^^7 = a, which gives the angular acceleration. 
at at' 

2. A flywheel is brought from rest up to a speed of 60 revolutions 
per minute in ^ minute. Find the average angular acceleration a, and 



98 DIFFERENTIAL CALCULUS 

the number of revolutions required. Find the velocity at the end of 
15 seconds. 

to = 60r.p.m. = 60 X ^ = 2 tt radians per sec. 
.-. at = a- SO = 2t or a = -J = 0.2094 rad./sec^. 

= 1 af2 = 1 1^ (30)2 = 15 X 27r = 15 revolutions. 

^ ^ ok) 

u = at = -^Xl5='ir = 3.14 rad. per sec. 

3. If the flywheel of Ex. 2 is 12 feet in diameter, find the tangential 
velocity and acceleration of a point on the rim. Find the normal 
acceleration at the instant full speed is attained. 

V = ro} = QX2ir = 37.7 ft. per sec. 
oct = ra = QX^ = QX 0.2094 = 1.256 ft./sec^. 



CHAPTER IV. 



GEOMETRICAL AND MECHANICAL APPLICATIONS. 



dy 
dx 



75. (a) Tangents and Normals. — Since the derivative 
= f {x) represents the slope of the curve y = / (a;) at any 



point (x, y), 

\-T-\ = slope of PiT = tan = mi, 

where is the angle XTQ, measured from the positive 
direction of the x-axis to the tangent TPi, and mi is the 
slope of the curve at the point Pi (xi, yi) . 

Hence from the equation of a line through a given point 
{^h Vi)} y — yi = 'in{x — Xi); the equation of the tangent 



at the point Pi (xi, yi) is y — yi 



Idxji 



(x — Xi), in which the 



subscript denotes that the quan- 
tity is taken with the value 
which it has at the point Pi. 
Since the sign of the derivative 
of a function indicates whether 
the function is increasing or 
decreasing, when mi is positive 
the curve is rising at Pi, and 
when mi is negative the curve 
is falHng there. If mi is zero 
the tangent is horizontal, parallel to, or coincident with the 
X-axis; and if mi is infinite, the tangent is vertical, parallel 
to, or coincident with ?/-axis. 

Points where the slope has any desired value can be found 

99 




100 DIFFERENTIAL CALCULUS 

by setting the derivative equal to the given number and 
solving the resulting equation for x. 

The slope of the normal NPi, being the negative reciprocal 
of the slope of the tangent TPi, is 

ni = = — cot = — ^ • 

Ml I dyji 

Hence, the equation of the normal is 

(6) Subtangents and Subnormals. — The subtangent 
and the subnormal are the projections on the a:-axis of the 
part of the tangent and normal, respectively, between the 
point of tangency and the a;-axis. 

From the figure: 

dx 
Subtangent TM = yi cot (j) = yi j- 

dy 
Subnormal MN = yi tan (j) = yi -^ 

Tangent TP, = Vmp' + tW = \Jy^' + y,' T^l' 

= 2/1 V 1 + 1 ^J^ = 2/1 cosec </). 

Normal NPi = Vmp' + mW = \J y^^ + 2/1^ 1"^? 



= ^Vl + [II = 2/isec0. 



If the subtangent is reckoned from the point T, and the 
subnormal from the point M, each will be positive or nega- 
tive according as it extends to the right or to the left. For 
any given curve the signs will depend upon the coordinates 
of the point of tangency. 

Note. — As mentioned before, the problem of tangents 
directly led to the Differential Calculus. 



ILLUSTRATIVE EXAMPLES 101 

76. Illustrative Examples. — 1. The circle x^ -{-y^ = a^. 
Differentiating, . 2 x dx -\- 2 y dy = 0, 

dy _ _x 
" dx y 

Idxji 2/1 

Equation of tangent, 

Xi , . 

y-yi= ~^/^~^i) 

or xxi + yyi = a?, after reducing. 
Equation of normal, 

y-yi = ^(^- ^0 

or yxi — yix = 0, after reducing. 

The final form of the last equation shows that the normal 
at any point on the circle passes through the center. 

The subtangent 

Idyji \ xj xi xi 

The subnormal MN = 2/1 T^l =yJ-^\= -xi. 

Since dy = dx, the ordinate of the circle changes 

y y 

times as fast as the abscissa; and since dy = da; is nega- 
tive, unless X and y have different signs, 2/ is a decreasing 
function of x in the first and third quadrants; while dy being 
positive when the moving point is generating the second and 
fourth quadrants, y is an increasing function of x in those 
quadrants. 

2. The parabola 2/^ = 2 px. 

Differentiating, 2ydy = 2pdx, 

, dy 
dx 



2, ' r^i = R. 

y " \_dxji 2/1 



102 DIFFERENTIAL CALCULUS 

Equation of tangent, y-yi = — (x — Xi) or yyi = p{x-\-Xi)j 
after reducing. 

Equation of normal, y — yi == —^(x — xi). 
The subtangent TM = yj^) = ^1 = ^^' =2xi. 
The subnormal MT = -yi (^^) = p. 

Hence, for any point on the parabola the subtangent is 
bisected at the vertex and the subnormal is constantly equal 
to p, the semi-latus rectum. These two characteristics of 
the parabola afford ready methods of accurately drawing a 
tangent at any point on the curve. 

Since dy = - dx, the rate oi y = - times rate of x. To 

^ y y 

find where the rates are the same, put -^ = - = 1, .*. y = p 

and X = ^ ; that is, the extremity of the latus rectum is the 

point where the rates of y and x are equal. Hence the tan- 
gents at the extremities of the latus rectum make angles of 
45° and 135° with x-axis and meet at right angles with each 
other at intersection of directrix and x-axis. It is evident 

diJi 
that at the origin where 2/ = 0, -p = oo ; that is, the 2/-axis 

is tangent at the vertex. It is seen also that, as y increases 
without limit, the tangent at its extremity becomes more 
and more nearly parallel to the x-axis. 

3. On the circle x^ -\- y'^ = \^io find the points where the 
slope is 1, 0, or 00 . 

[11 =-:-:-■■ ■■■ »■=-- 



ILLUSTRATIVE EXMAPLES 103 

substituting in x^ + 2/' = 1, Xi = =hj V2 and i/i = T^ V2. 

W = _ El = 0, /. a;i = and y, = ±1, 
L«^Ji 2/1 

by substituting in x^ + 2/^ = 1. 

[3=-l="' •■• ^'=° ^"^ ^^=^^' 

by substituting m x^ -\- y^ = 1. 

4. To find at what angle the circle x^ + y^ = 8 and the 
parabola y"^ = 2x intersect. 

Making the two equations simultaneous, the points of 
intersection are found to be (2, 2) and (2, — 2) . 

xi 2 



For circle, mi = 
For parabola, mi = 



dx_ 
'dy 
dx 



£ = J- 

2/1 ±2 



For angle of intersection, 

_ J l_ 

tan 4> = — = 1 = t3, 

1 + mim2 1 — 2 

or = tan~i (=F3), and from table of tangents, = 108° 26' 
or 71° 34'. 

5. The path of a point is the arc of a parabola y^ — 2 px, 
and its velocity is v; find its velocity parallel to each axis. 

Let s denote the length of the path measured from any 

point on it; then -ri = v. 

From 2/^ = 2 px, 

dy _p dx 

dt y dt 
Substituting these values in 

e)"=(S)'+(S)" <"■")■ 



104 DIFFERENTIAL CALCULUS 



tdxV- _ y^ _ yV dx _ yv 



dy p dx pv 



dt y dt -y^yi -j- p2 

6. A comet's orbit is a parabola, and its velocity is v; find 
its rate of approach to the sun, which is at the focus of its 
orbit. 

Let p denote the distance from the focus to any point on 
2/2 = 2px; then p = x -\- ^ p, from point to directrix; 

dp _ dx 
" di~di' 

p being constant. Hence, the comet approaches or recedes 
from the sun just as fast as it moves parallel to the axis of 
its orbit; 

dp dx y 



dt dt \/y2 _j_ p2 



V. (Example 5.) 



At the vertex, y = 0; hence, at the vertex -^ is zero. When 



y = v, 



dp dx ^ . 2/^1 

-jT = -jT <v, smce / < 1. 

dt dt V 2/2 + p2 

^^- ^' (Examples) 



dt \/i/2 +p2' 

pv 

fdsV ^ (dxV (d^V 
\dt) \dt)~^\dt/ 

la 

limit as y increases without limit. 



and lim , = o • 

y=00 Vi/2 + p2 ' 

dx , ds 

■'■ -dt^dt^"' '''"'' 

Hence -^ = 777 is always less than v and approaches v as a 

CLt O/l 



ILLUSTRATIVE EXAMPLES 



105 



7. To compare the velocity of a train moving along a 
horizontal tangent with the velocity of a point on the flange 
of one of the wheels, and to compare also the horizontal and 
vertical components of the flange point. 

Let a wheel whose radius is a roll along a horizontal line 
with a velocity v; find the velocity of any point P on its rim, 
also the velocity of P horizontally and vertically. 




(1) 



OM D X 

The path of P is a cycloid whose equations are: 
X = a{6 — mid), ) 

y = a{l — cos 6) = a vers d, ) 

where 6 denotes the variable angle DCP, and a the radius 
CD. 

Since the center of the wheel is vertically over D, 

V = the time-rate of OD 



(2) 





d(ad) 
dt 


= a 


dd 
'dt 


dd 
dt 


_ V 

a 







Differentiating equations (1) gives, by (2), 

dx .. ^.dd .dd . 

-TT = a (1 — cos ^) -r: = a vers d-rr- = v vers d 
dt ^ ^ dt dt 



and 



= the velocity horizontally, 
-r: = asind -7- = v sin d = the velocity vertically. 



(3) 
(4) 



106 DIFFERENTIAL CALCULUS 



= velocity of P along its path. (5) 

The velocity of P may be considered as the resultant of two 
velocities each = v, one along PT tangent to the circle and 
the other along PH parallel to the path of C. The resultant 
PB must bisect the angle HPT; :. DPB = 90° and PB is 
tangent to the cycloid, the path of P, making PD the 



normal. 






At 0, d = 0, 


and 


dx dy ds 
dt~ dt'di'^- 


AtP, = |, 




dx 1 dy 1 /- ds 

di = r' 1 = 2'^^' di 


AtP,,d = '^, 




dt dt ' dt 


At P2, 61 = TT, 




^-ds_ dy _ 
dt'dt-"^'' dt-^' 


From (5) is obtained 






ds 
dt 






: V = V2a'y : a. 



-rr = «^. 



Hence, the velocity of P is to that of C as the chord DP is 
to the radius DC; that is, P and C are momentarily moving 
about D with equal angular velocities. (See Art. 72.) 

When 6 = 60°, their linear velocities also are equal, as 
shown above. 

8. Find the equation of the tangent and the values of the 
subnormal and normal of the cycloid. 

Dividing (4) by (3), Example 7, gives 



sin^ 


V(2 


a — 1 


}j)y/a 


vers0 
PH 


^HB' 


DH 


and 



dy _ &md _ V{2a - y)y/a ^ . / (2 a - y) 
dx vers 6 y/a V y 



since sin 6 = -p^ = and vers d =- from (1) ; 

CP a a 



EXERCISE IX 107 

is the equa,tion of the tangent at point (xi, yi). 

n^i 1 T dy sin^ sin^ . ^ „^, ,,t^ 

The subnormal = 2/ 3^ =y n=y—T- =asme = PH = MD. 

ax vers d y/a 

Thus the normal at P passes through the foot of the per- 
pendicular to OX from C. Hence, to draw a tangent and 
normal at P, locate C, draw the perpendicular DCB equal 
to 2 a, and join P with B and D; then PB and PD will be 
respectively the tangent and normal at P. 

Normal = DP = VDB • DH = V2a^y. 

9. Eliminating d in equations (1) of Example 7, equation 
of cycloid is 

X = tt' arc vers ?//a =F V2 ai/ — ?/^, 
since B = arc vers y/a and a • sin = =b V(2 a — i/) a. 

EXERCISE IX. 

Deduce the following equations of the tangent and the normal: 

1. The ellipse, x^/a^ + yyh^ = 1, Xix/a^ + yiy/b^ = 1, 

2. The hyperbola, x^a^ - ?/V62 = 1, xix/o? - 2/12//&2 = 1 



— /727 



3. The hyperbola, 2xy = a^, Xiy + ^/ix = a^, ?/i?/ — 0:1.1; = yi^ — Xi^. 

4. The circle, x^ -\- y^ = 2 ax, y — yi = (x — Xi) {a — Xi)/yi, 

y - yi = {x - xi) yi/{xi - r). 

5. Find the equations of the tangent and normal at (3/2 a, 3/2 a): 
a^ + y^ = S axy. ^ Ans. x -\- y = Za, x=y. 

6. x + y = 2e='-y,Sit (1, 1). 

Ans. 3y = x-\-2, Sx + y = 4. 

7. ix/ar + (y/br=^2,at(a,b). 

Ans. x/a + 2//& = 2, ax — by = a^ — ¥. 



108 DIFFERENTIAL CALCULUS 

8. Show that the sum of the intercepts of the tangent to the para- 
bola X* + 2/^ = a^, is equal to a. 

9. Show that the area of the triangle intercepted from the co- 
ordinate axes by the tangent to the hyperbola, 2 xy = a?, is equal to o?. 

10. Show that the part of the tangent to the hypocycloid x^ -\-y^ = a% 
intercepted between the axes, is equal to a. 

11. Find the slope of the logarithmic curve x = logb y. The slope 
varies as what ? What is the slope of the curve x = logy? 

12. Find the normal, subnormal, tangent, and subtangent of the 
catenary y = a/2 {e^^"' + e-^/*"). 



Vy^ - a2 V 2/2 - a2 
13. At what angles does the line Sy — 2x — 8 = cut the parabola 
= 8 X? Ans. arc tan 0.2; arc tan 0.125. 

77. Polar Subtangent, Subnormal, Tangent, Normal. — 

Let arc mP = s, and arc PQ = As; then z POQ = Ad, 




T 

circular arc PM = pA0, and MQ =- Ap. The chords PM 
and PQ, the tangents RPU and TPZ, are drawn; and ZR 
is drawn perpendicular to PR, Z being any point on the 
tangent PZ. 

When As = 0, the hmiting positions of the secants PM 
and PQ are the tangents RPR and TPZ, respectively; hence, 
li ( z PMQ) = z RPK = 7r/2 = z PRZ, 
Itiz OQP) = z OPT = tA = Z RZP, 
and It z MPQ = zRPZ. 



POLAR SUBTANGENT 109 

Now in a problem of limits the chord of an infinitesimal arc 
can be substituted for the arc, since the limit of their ratio is 
unity (Art. 22 and Cor., Art. 46) ; so 

Ap^ MQ ^ sin MPQ ^ 

As chord PQ sinPMQ' 

, . , nA(9 , chord MP , sinMQP 

Agam, It -r — = It -j T ^^ = tt - — r^T.^^ ; 

^ ' As chord PQ smPMQ' 

From (1) and (2), it follows that, if PZ is taken as ds, 
ds = PZ, dp = HZ, and pdd = HP. 

Drawing OT perpendicular to OP, and PA and ON perpen- 
dicular to the tangent TP, the length PT is the polar tan- 
gent; PA, the polar normal; OA, the polar subnormal; and 
OT, the polar suhtangent. 
From the right-angled HPZ 

ds'' = dp''-{-p^dd^; (3) 

• , pdd , dp , , pdd ,.. 

sm^ = -^, cos^ = ^, tan^ = ^. (4) 

Polar subt. = OT = OP tan xp = p^ dd/dp. (5) 

Polar subn. = OA = OP cot xp = dp/dd. (6) 

Polar tan. = PT = VOP' + OT' = p y/n- p2 ^ . (7) 
Polar norm. = AP =- VOP' + OA' = y p' + ^- (8) 



p = ON = OP sinxp = p2 d6>/ds 
P^ 



(9) 



Vp2 + (dp/dey 

<i> = yP-\-e. (10) 

Corollary. — If PZ represents the velocity at P of a moving 



110 DIFFERENTIAL CALCULUS 

point (p, e) along its path, PK (= HZ) and PH will repre- 
sent its component velocities at P along the radius vector 
and a line perpendicular to it. 

If the path is a circle with center at 0, \p is 90°; and 

I • f\r\o 1 pdd J. ... ds dd 
smi/' — sin90 = 1 = -7-, Irom (4), or -^ = p-^-? /. v = rco; 

that is, the linear velocity = radius times angular velocity. 
(See Art. 72.) 

EXERCISE X. 

1. Find the subtangent, subnormal, tangent, normal, and p of the 
spiral of Archimedes p = ad. 

Ans. subt. = p^/a; subn. = a; normal = Vp2 -|- a^; 
tangent = p VF+TVo^; P = P^lip^ + a^)^ 

2. In the spiral of Archimedes show that tan ^ = d] thence find the 
values of \p, when = 2 tt and 4 tt. 

Ans. 80° 57' and 85° 27'. 

3. Find the subtangent, subnormal, tangent, and normal of the 
logarithmic spiral p = a^. 

Ans. subt. = p/loga; subn. = ploga; tan. = p Vl -f- (loga)"^; 

norm. = p Vl + (log ay. 

4. Show why the logarithmic spiral is called the equiangular spiral, 
by finding that yp is constant. 

li a = e, -^ = 7r/4, subt. = subn., and tan. = norm. 

5. Find the subtangent, subnormal, and p of the Lemniscate of 
Bernouilli p^ = a^ cos 2 6. 

Ans. subt. = — pV«^ sin 2 d] subn. = —a^ sin 2 d/p] 
p = pyVp^ + a^ 8111^2 6 = p^la^, 

6. In the circle p = a sin 0, find ^ and 0. 

Ans. i/- = 0, and <}> = 2 0. 
The angle between two polar curves is found as for the other curves. 

7. Find the angle of intersection between the circle p = 2 a cos 6, 
and the cissoid p = 2 a sin tan 6. 

Ans. arc tan 2. 



CHAPTER V. 
MAXIMA AND MINIMA. INFLEXION POINTS. 

78. Maxima and Minima. — One of the principal uses 
of derivatives is to find out under what conditions the value 
of the function differentiated becomes a maximum or a 
minimum. 

This is often very important in engineering questions, 
when it is most desirable to know what conditions will make 
the cost of labor and material a minimum, or will make 
efficiency and output a maximum. 

A maximum value of a function or variable is defined to 
be a value greater than those values immediately before and 
after it, and a minimum value to be one less than those 
immediately before and after it. It follows that the function 
is increasing before, and decreasing after reaching a maxi- 
mum value; while it is decreasing before, and increasing 
after reaching a minimum value. 

The points on the graph oiy = f {x) at which the function 
ceases to increase and begins to decrease, or ceases to de- 
crease and begins to increase, are maxima or minima points; 
and the values of the function at those points are maxima 
or minima values. 

It is to be noted that a maximum value is not necessarily 
the greatest value the function can have nor a minimum the 
least; f (a) is a maximum if it be greater than any other value 
of / (x) near / (a) and on either side of it ; and / (a) is a mini- 
mum if it be less than any other value of / (x) near / (a) and 
on either side of it. 

79. The Condition for a Maximum or a Minimum Value. 
— If / {x) is a function of an increasing variable x; then 

111 ' 



112 DIFFERENTIAL CALCULUS 

for / (a) to be a maximum, / (x) must be increasing just 
bef ore / (a) and therefore / (a:) must be positive; on the 
other hand / (x) must be decreasing just after / (a) and 
therefore f (x) must be negative. Hence, as x increases 
through the value a, f (x) must change from a positive to a 
negative value. Conversely, if as x increases through the 
value a, f (x) changes from a positive to a negative value, 
/ (a) will be a maximum value of / (x) . 

Hence / (a) will be a maximum value of / (x) if, and only 
if, / (x) changes from a positive to a negative value as x 
increases through the value a. 

In the same way it may be seen that/ (a) will be a minimum 
value of / (x) if, and only if, f (x) changes from a negative to 
a positive value as x increases through the value a. 

This condition has been called the fundamental condition 
or test. 

For the cases of most frequent occurrence; when f (a) is a 
maximum or a minimum, f (a) = 0. In most cases it is a 
necessary condition for a maximum or a minimum value of 
a function that the first derivative at that value shall be 
zero. For in most cases the first derivative /' (x) is con- 
tinuous; and, when continuous; it changes sign by passing 
through the value zero only. But if f (x) is not continuous, 
as is the case for some functions, then it may change sign by 
becoming infinite for some finite value of x; for if /' (x) is 
a fraction whose denominator becomes zero for some finite 
value of X, f (x) changes sign as x increases through that 

value. For example, when f'(x) = -, ior x = a = 2, 

X z 

f (^) = o o ^ ^ ' ^^^^ f (^) ^^ negative before, and 

positive after x increases through the value 2; hence, / (2) 
is a minimum according to the fundamental test. Again, 
there are exceptional non-algebraic functions for which/' (x), 
as X increases through some finite value a, changes sign 



GRAPHICAL ILLUSTRATION 



113 



without becoming either zero or infinite. (See Note, 
Art. 80.) 

Excepting such rare functions, a theorem may be stated 
thus : 

For all algebraic functions any value of x which makes f {x) 
a maximum or a minimum is a root of f (x) = or f (x) = cc . 

The converse of this theorem is not true ; that is, any root 
of /' (x) = or f (x) = 00 does not necessarily make / (x) 
either a maximum or a minimum. These roots are called 
critical values of x, and each root may be tested by rule. 

80. Graphical Illustration. — Let P . . . P3 . . . Pt be 
the locus oi y = f (x). Then / (x) will be represented by the 
ordinate of the point (x, y), and /' (x) by the slope of the 
locus at the point (x, y). By definition, the ordinates MP, 
M2P2, and M4P4 represent maxima of / (x) ; while 0, MiPi, 
M3P3, and M5P5 represent minima. (Art. 78.) 




The slope f (x) is positive immediately before a maximum 
ordinate, and negative immediately after; while the slope is 
negative immediately before a minimum ordinate and posi- 
tive after. The slope / (x) is or 00 at any point whose 
ordinate / (x) is either a maximum or a minimum. The 
slope / (x) is discontinuous at the points P4 and P5, where 
it changes sign by becoming infinite as x increases through 
the values OM4 and OM5; that is, / (a;) = 00 . 




114 DIFFERENTIAL CALCULUS 

The slope f {x) is at Pe and oo at P7; but it does not 
change sign at either point, and neither M&P^ nor M^P^ is a 
maximum or a minimum ordinate; it does, however, change 
in value at each point, Pe being a point where the slope f (x) is 
a minimum and P7 one where it is a maximum. The points 
Pe and P? are inflexion points, at which the curve changes 
from being concave downward to upward, or vice versa. 

Note. — Points such as P4 and P5 occur on railroad '^ Y's," 
and such points where branches of a curve end tangent to 

each other are called cusps. 
At a point on a non-algebraic 
curve where branches end and 
are not tangent to each other, 
called a shooting point, f (x) 
may change abruptly from a 
positive finite value to a nega- 
tive value, or vice versa; hence, 
/ (a) would be a maximum or a minimum without / (x) 
becoming either zero or infinite. The supplementary figure 
shows a shooting point at which / (a) is a minimum; / (x) 
becoming — 1 as x increases to a, and +1 as x decreases to 
the same value a, thus changing from a negative to a positive 
value as x increases through the value a. 

It is to be noted that, while on an exceptional curve like 
the one shown the tangents at a maximum or a minimum 
point may have various directions, on any algebraic curve 
the tangent is parallel to one or other of the two rectangular 
axes; that is, the tangent at a maximum or a minimum point 
is horizontal, the slope being continuous; otherwise it is 
vertical ; and on only exceptional non-algebraic curves will it 
have any other direction. 

It may be noted also, as in the graphical illustration 
given, that maxima and minima occur alternately; that is, 
a minimum between any two consecutive maxima and vice 
versa. It may be seen that a maximum may be less than 



RULE FOR APPLYING FUNDAMENTAL TEST 115 

some minimum not consecutive, since by definition it is 
necessarily greater than those values only immediately before 
and after it. It may be seen also that when the slope is 
continuous at least one inflexion point must occur between 
a maximum and a minimum point. The only inflexion 
points marked on the curve are Pe and P7, occurring where 
f (x) = and 00 , but f (x) may have any value at an 
inflexion point, although its rate, f (x), must change sign 
there, becoming or QO . Hence, at any inflexion point, a 
point where the slope f (x) is a maximum or a minimum, 
/' (a:) = or QO . The converse is not true, for /" (x) may 
be or 00 at other points. 

81. Rule for Applying Fundamental Test. — Let a be a 
critical value given by either f (x) = or f (x) = 00 , or, in 
general, any value of x to be tested, and Ax a small positive 
number; then: 

If f {'^ ~ ^^) ^'s positive and f {a + Ax) is negative, 

f (a) is a maximum off (x) . (Art. 79.) 

Uf iP' ~ Ax) is negative andf (a + Ax) is positive, 

f (a) is a minimum of f (x). (Art. 79.) 

If f (^ ~ Ax) and / (a + Ax) are both positive or both 

negative, f (a) is neither a maximum nor a minimum of 

This rule is general and is valid for all functions that are 
continuous one-valued functions, which comprise all those 
usually encountered in this connection. 

82. While the rule just stated applies in every case; 
when /' (x), as well as / (x), is continuous and therefore the 
critical values of x are roots of/' (x) = 0, a rule usually easier 
to apply may be deduced from the fundamental test or 
condition. 

Let a be a critical value of x given by /' (x) = 0. If / (a) 
is a maximum value of / (x), /' (x) changes from a positive 
to a negative value as x increases through a; therefore, near 
a,f' (x) is a decreasing function, and therefore, its derivative, 



116 DIFFERENTIAL CALCULUS 

f" {x), must be negative near a. But if f" (a) is not zero, 
then near a the sign of f (x) is that of /' (a). Hence 
f (a), if it is not zero, will be negative when/ (a) is a maxi- 
mum value of/ (x). 

In the same way it is seen that/' (a), if it is not zero, will 
be positive when / (a) is a minimum value of / (x) . 

Conversely, / (a) will be a maximum or a minimum value 
of / (x) according as /" (a) is negative or positive. 

Hence this rule for determining the maxima and minima 
values of/ (x) when/ (x), f (x) are continuous: 

The roots of the equation f (x) = are, in most cases, the 
values of x which make f (x) a maximum or a minimum. 

If a he a root of f (x) = 0; then f (a) will he a maximum 
value off (x), if f" (a) is negative, hut a minimum, if f" (a) is 
'positive. 

83. While the above rule is all that is needed in most 
cases, it does not provide for the case when the critical value 
a makes/'' {x) become zero. When/' (a) = 0,/ (a) may be 
either a maxinium or a minimum, or it may be neither, and 
the point on the graph of / {x) may, or may not, be a point of 
inflexion; so an extension of the rule is needed to provide for 
cases where /" {x) and the succeeding derivatives may in 
turn become zero for the value, a. 

If no derivative is found that does not become zero when 
a is substituted for x, then recourse may be had to the funda- 
mental test, that rule applying in every case. But if/' (a), 
/" (a), . . . , /"""^ (a) all are found to be zero, and /"(a) not 
zero; then the following rule, inclusive of the preceding, 
applies. Let a he a critical value of x given hy /' (x) = 0, 
and let a he substituted for x in the successive derivatives 

of f (x). 

If the order n of the first of the derivatives that is not zero is 
an even integer, f (a) will he a maximum or a minimum off (x) 
according as this derivative is negative or positive. 

If the order n of the first of the derivatives that is not zero is 



RULE FOR DETERMINING MAXIMA AND MINIMA 117 

an odd integer, f (a) will be neither a maximum nor a minimum 
off (x) regardless of the sign of this derivative. 

Note. — This conclusion can be deduced by examining 
the signs of the derivatives near a; thus, as follows: 

If /' (a) and f' (a) be zero but f" (a) not zero; since 
/'" {x)j the rate of f^ (x), has when x is a a value not zero; 
/" (x), the rate of /' (x), is then increasing or decreasing 
according as /''' (a) is positive or negative, and, since it is 
zero when x is a, it must change sign as x increases through a ; 
therefore, f (x), the rate of / (x), must be either decreasing 
before and increasing after, or increasing before and decreas- 
ing after x is a, and so, continuing to be positive or negative 
according as f" (a) is positive or negative, does not change 
sign as x increases through a; hence / (a) is neither a maxi- 
mum nor a minimum of/ (x) regardless of the sign oi f" (a). 

Nowif /'' (a) also is zero but /^^ (a) not zero; since /^ (x), 
the rate oi f" (x), has when x is a a value not zero, /"' (x), 
the rate of /" (x) , is then decreasing or increasing according 
as /^^ (x) is negative or positive and, since it is zero when x 
is a, it must change sign as x increases through a; therefore, 
f (x), the rate of/' (x), must be either increasing before and 
decreasing after, or decreasing before and increasing after x 
is a, and so, continuing negative or positive according as 
p^ (a) is negative or positive, does not change sign as x 
increases through a; f (x), the rate of / (x), must then be 
either decreasing, or increasing before and after x is a, and, 
as it is zero when x is a, it changes from a positive to a nega- 
tive value, or from a negative to a positive value, as x in- 
creases through a, according as /^^ (a) is negative or positive; 
hence / (a) is a maximum or a minimum of / (x) according 
as /^^ (a) , the first of the derivatives that is not zero, is neg- 
ative or positive. 

In the same way it follows that, if /^ (a) is the first of the 
derivatives that is not zero, / (a) is neither a maximum nor 
a minimum of / (x) regardless of the sign of /^ (a) ; and that, 



118 



DIFFERENTIAL CALCULUS 



if /^^ (a) is the first, / (a) is a maximum or a minimum of/ (x) 
according as /^^ (a) is negative or positive ; and so on for the 
succeeding derivatives: hence the inclusive rule given. 
(For proof by Taylor's Theorem, see Art. 218.) 

f(a)=f(oJ=o, neither a 7ruix.noraminoff(x) f(a) =f(o)=o, a mm. of (x) 

r 




-25"- 
y=f(x)=-4x^ 
f(a) ^£i[oJ=o,,mithera rruix^. mra W-itl^ o£f(x) 



-25-L 



y=f(x)=-x4 
ffa) -MJ ^o,a max. off(x) 



84. Typical Illustrations. — The foregoing deductions 
may be verified by the graphs of the successive derivatives 
of x^^ — x^ xS and — x^ referred to the same axes as those of 
the graphs of the functions. The usual case when f (a) is 



TYPICAL ILLUSTRATIONS 



119 



zero and /" (a) not zero, is well illustrated by the graphs of 
the function sin 6 and its derivatives. 



f(ej=sinf-) 



f'(w<ose 




, iir/2 + nTr) 



neh-sine 



f{d) = sin^; where ^ is in radians. 
f'{d) = cos0 = 0; .-. d = 7r/2, Itt, . 
r{d) = -sine, r(j/2)= -; 

.-. / (7r/2) = 1 is a maximum value of / (x). 
J'' (3 ^) = _|- . /. / (I tt) = - 1 is a minimum value of / (x) . 
j" \e) = - sin^ = 0; .*. ^ = 0,7r, . . ., mr] 

:. (0, 0), (tt, 0), . . . are points of inflexion. 

f"{e) = -cosd; r{o) = -; 

/. f (o) = 1 is a maximum slope. 
jf'f (^^) ^ _|_ . ^.^ f (^) = _ 1 is a minimum slope. 

The graphs make manifest that for a maximum or a mini- 
mum value of the function the first derivative passes through 
zero; being + before and - after for a max., and - before and 



120 DIFFERENTIAL CALCULUS 

+ after for a min.; and that hence the second derivative is 
— for a max. and + for a min. ; also that at an inflexion point 
the second derivative is zero. The graph of sin makes 
manifest at once the maxima and minima values of the 
function and the value of the angle in radians that makes the 
function a maximum or a minimmn. 

. The graph of the first derivative shows that the first 
derivative of any continuous function when continuous itself, 
changes sign by passing through zero only, for the ordinate 
changes sign by becoming zero only, as the graph crosses the 
axis of &; and it shows that in passing through zero the 
ordinate changes from plus to minus, or from minus to plus 
according as the abscissa of the point of crossing corresponds 
to a maximum or a minimum ordinate of the graph of the 
function ; and that as it crosses the axis the ordinate is either 
decreasing or increasing. The graph of the second derivative 
shows by the direction or sign of its ordinate at or near the 
point where the graph of the first derivative crosses the axis 
whether the first derivative is decreasing or increasing as it 
passes through zero at that point, the sign being minus or 
plus according as the first derivative is decreasing or increas- 
ing; that is, according as the abscissa of the point corre- 
sponds to a maximum or a minimum ordinate of the graph 
of the function. 

85. Inflexion Points. — Where the slop&.oi the graph of 
the function is a maximum or a minimum is at the points 
where the ordinate of the first derivative has its greatest 
positive or negative value, and those points are precisely 
where the ordinate of the second derivative is zero, those 
values of 6 that make the slope of the function a maximum 
or a minimum being those that make its derivative, the 
second, zero. These points are inflexion points on the graph 
of the function, where the curve changes from being concave 
upward to downward, or downward to upward. When a 
curve as mn is concave upward its slope evidently increases 



INFLEXION POINTS 



121 



as the abscissa of the generating point increases; hence its 
derivative, the second (the flexion), is positive. When a 
curve as st is concave downward its slope evidently decreases 
as the abscissa increases; hence the second derivative is 
negative. At a point of inflexion, as P on mt or sn, the tan- 
gent crosses the curve at the point of contact, and on opposite 






sides the curve is concave in opposite directions, therefore, 
the second derivative has opposite signs. Hence, at a point 
of inflexion the first derivative, the slope of the curve, has a 
maximum or a minimum value. To test a curve for points 
of inflexion is to test its slope for maxima and minima. In 
the case of roads or paths that change direction of curvature 
in a horizontal plane the inflexion point is usually called the 
point of reverse curvature, and when the curved grade 
changes direction of curvature in a vertical plane the inflexion 
point is where the grade is greatest or least on that part of 
the road. The roots of the second derivative = or oo are 
the critical values to be tested for points of inflexion, and 
the sign of the third derivative, when the critical value is 
given by the second derivative = 0, indicates whether the 
second derivative, the flexion, is decreasing or increasing as 
the critical value is passed, the sign being minus or plus 



122 



DIFFERENTIAL CALCULUS 



according as the second derivative is decreasing or increas- 
ing; that is, according as at the critical value the slope is a 
maximum or a minimum, or as the curve is concave upward 
before and downward after or the reverse. These conclu- 
sions are verified by the graphs of sin $ and its successive 
derivatives. 

When the critical value is given by the second derivative 
= 00, or, in general, when any value is to be tested, the 
fundamental test may be appUed to determine the sign of 
the second derivative before and after the value to be tested, 
and thus to determine the concavity and existence or non- 
existence of inflexion. 

It is evident that at a maximum point on a curve the 
curve is concave downward both sides of the horizontal 
tangent point and concave upward both sides of a vertical 
tangent point, while the reverse is the case at a minimum 
point. 




Tangents drawn at successive points on a curve that has 
maxima and minima points show that as the abscissa of the 
moving point increases the tangent turns clockwise through 
zero angle at a maximum point until an inflexion point is 
reached when it <^urns in the opposite direction through zero 
angle again at a minimum point; and then it may without 
any inflexion point turn through a right angle at a maximum 
point, continuing to turn anti-clockwise until possibly at an 
inflexion point it turns back clockwise through a minimum 
point when the angle is a right angle again, becoming less as 



POLAR CURVES 



123 



the tangent continues to turn clockwise. The points on a 
graph at which the ordinate ceases to increase and begins to 
decrease, or else ceases to decrease and begins to increase are 
sometimes called turning points of the graph, and the corre- 
sponding values of the function turning values. The turning 
values are evidently maxima and minima values Eind the 
turning points maxima and minima points. While the tan- 
gent at an inflexion point turns in opposite directions, the 
curve is either rising or falling on both sides of the point; but 
at a turning point the curve is rising at a maximum and then 
falling, or falling at a minimum and then rising. These 
considerations make it obvious that at a maximum, the 
angle made by the tangent decreasing, its rate, the second 
derivative of the function, is negative, and that at a mini- 
mum, the angle increasing, the second derivative is positive. 




86. Polar Curves. — A polar curve is concave or convex 
to the pole at a point, according as the tangent to the curve 
at the point does not, or does lie on the same side of the 
curve as the pole. It may be seen from the figure that 
when a polar curve, as mn, is concave to the pole, p or ON 
increases as p increases; hence, the rate of change of p with 

respect to p, -^ is positive. 
dp 

When a curve, as st, is convex to the pole, p decreases as 

p increases; hence, 3^ is negative. 
dp 



124 DIFFERENTIAL CALCULUS 

It follows that a polar curve is concave or convex to the pole 

at a point according as ~ is positive or negative. 

dv 
At a point of inflexion on a polar curve, as P on mt, -J- 

dp 

changes sign, and therefore p is Si maximum or a minimum; 

and conversely. Hence to test a polar curve for points of 

inflexion, p is tested for maxima and minima. 

Example. — Examine the Lituus for points of inflexion. 
H P^ ^ 2a^p , 

^^^ ^ Vp2 + {dp/ddy V4a4 + p4' 

-r- — 5 — — ^, " p = a\^ z. 

dp (4a4 4-p4)f 

Hence, p = a a/2 makes p a maximum; and (a V2, j) is a 
point of inflexion. 

d7) 
The spiral p = a^ has no point of inflexion, since -—■ is 

dp 

always positive. 

87. Auxiliary Theorems. — By use of the following 
theorems, which are obvious, the solutions of problems in 
maxima and minima are often simplifled : 

(i) Any value of x which makes c + / (x) a maximum or a 
minimum makes / (x) a maximum or a minimum; and 
conversely. 

(ii) Any value of x which makes c»f{x), c being positive, 
a maximum or a minimum makes / (x) a maximum or a 
minimum; and conversely. If c is negative and / (a) is a 
maximum, c •/ (a) is a minimum. 

(iii) Any value of x which makes f{x) positive, and a 
maximum or a minimum, makes [/ (x)]'' a maximum or a 
minimum, n being any positive whole number. 

(iv) Since / (x) and log / (x) increase and decrease to- 
gether, any value which makes / (x) a maximum or a mini- 
mum makes log/ (x) a maximum or a minimum; and con- 
versely. 



EXERCISE XI 125 

(v) Since when / (x) increases its reciprocal decreases, any 
value of X which makes / (x) a maximum or a minimum 
makes its reciprocal a minimum or a maximum. 

EXERCISE XI. 

Examine / (x) for maxima and minima when: 

1. fix) =0^5-50^ + 5x3-1. 

fix) = 5x4 -20o;3-l- 15x2 = 5a;2(a.2_4^ + 3) 

= 5 x2 (x - 1) (x - 3) = 0; .-. X = 0, 1, 3. 
f (x) = 20 x3 - 60 x2 + 30 x; /'" (x) = 60 x^ - 120 x + 30. 
.^ _ r(0)=0, r'(0) =30; .-. /(O) = -1 
is neither a max. nor a min. 

r (1) = 20 - 60 + 30 = -10; .-. / (1) = is a max. 
f (3) = 540 - 540 + 90 = 90; .'. / (3) = -28 is a min. 
By plotting the graph of / (x) these results may be verified. 

2. / (x) = x3 - 3 x2 + 3 X + 7. 

/' (x) = 3 x2 - 6 X + 3 = 3 (x2 - 2 X + 1) 

= 3 (x - 1)2 = 0; .-. X = 1, 1. 
f ' (x) = 6x - 6 = 6 (x - 1); f " (x) = 6. 
f (1) = 0;/'"(l) = 6; .-. /(I) = 8 isneither amax. noramin. 

3. /(x)= 3x4 -4x3 + 1. 

/' (x) = 12 x3 - 12 x2 = 12 x2 (x - 1) =0; .'. x = 0, 1. 
/" (x) = 36x2 _ 24x = 12x (3x - 2); /'" (x) = 72x - 24. 
r(0)=0;r'(0) = -24; .-. /(O) =1 
is neither a max. nor a min. 

f'(l)=12; .-. / (1) = is a min. 

4. fix) =3x5- 125x3 + 2160X. 

/' (x) = 15x4 - 375x2 + 2160 = 15 (x^ - 25x2 + 144) = 0; 

.'. X = ±3, ±4. 
f (x) = 15 (4 x3 - 50 x) ; /" (3) is neg. ; .-. / (3) is max. 
f (4) is positive; .*. / (4) is min.; /" (-3) is positive; .'. / (-3) 
is min.; /" ( — 4) is negative; .'. / ( — 4) is max. 

5. fix) =x3-3x2 + 6x + 7. 

f (x)=3a;2-6x + 6 = 3(x2-2x + 2) = 0; .'. x = l±V3i. 
Hence no real value of x makes / (x) a max. or a min. 

6. Examine c + V4 a2x2 — 2 ax^ for max. and min. 
Let fix) =2 ax2 - x^ (by Art. 87). 

/'(x) = 4ax-3x2 =4(4a - 3x) =0; .'. x = 4/3a,0. 
f" (x) = 4a - 6x; /" (0) = 4a; .'. /(O) = c is a min. 
/"(4/3a) =4a-8a = -4a; 

.-. / (4/3 a) = c + 8 a2 \/3/9 is a max. 



126 DIFFERENTIAL CALCULUS 

7. y = a — S{x — c)^, and y = a — h {x — c)^. 

J- = 1 = 00 ; /. X = c, and it can be seen that -r 

dx S (x - cy dx 

changes from + to — when x passes through the value c, hence/ (c) = a 

is a max. 

When/(a:) =a-hix-c)y, £= -— -^ = oo ; /. x = c; 

"^ S{x — cy 

dv 
here it can be seen that /' (x) or -7- does not change sign as x passes 

through c, and, therefore, the function has neither max. nor min. 

8. Examine {x — 1)* (x + 2)^ for max. and min. 

fix) = (x-iy(x + 2)H7x-i-5)=0; .: a: = 1, -2, -f 
/' (1 - Ax) is -, f (1 + Ax) is +; .-. / (1) = is a min. 
/'(-f -Ax) is +,/'(-f +Ax) is -; .-./(-f) is a max. 
/' (-2 - Ax) and /' (-2 + Ax) are both +; hence / (-2) is 
neither max. nor min. 

9. Examine ^ — for max. and min. 

a — 2x 

f (x) = {a - x)H4:X - a)/{a - 2 x)^; f {x) = gives x = a, o/4; 

/' {x) = 00 gives (a — 2 xy = 0, or a: = a/2. 

f (a/4) changes from — to + as a; passes through a/4; .*. f(a/i) 
is min. When x = a, or a/2, f (x) does not change sign; .*. f{a) and 
/ (a/2) are neither max. nor min. 
/p2 7 a; _!_ g 

10. Examine z-p: — for maxima and minima. 

a; — 10 

Ans. f (4) is a max.; / (16) is a min. 

(x + 2)^ 

11. Examine ) :^„ for maxima and minima. 

[x — 6)^ 

Ans. f (3) is a max; / (13) is a min. 

12. When fix) = (x - 1) (x - 2) (x - 3), / (2 - I/V3) = f V3, 
is a max., and/ (2 + 1/^3 ) = — f Vs is a min. 

13. Show that the maximum value of sin 6 + cos 6 is V2. 

14. Show that the maximimoi value of a sin + 6 cos 6 is Va^ -}- 6^. 

15. Show that e is a minimum of x/log x. 

16. Show that 1/ne is a maximum of log x/x'^. 

17. Show that e^^^ is a maximum of x^^. 

18. Show that 1 is a maximum of 2 tan ^ — tan^ d. 

19. Find the maximum value of tan"^ x — tan"^ x/4, the angles 
being taken in the first quadrant. 

Ans. tan~^ f . 

20. Show that 2 is a maximum ordinate and —26 is a minimum 
ordinate of the curve y = x^ — 5x*-^5x^-\-l. 



EXERCISE XI 



127 



PROBLEMS IN MAXIMA AND MINIMA. 

1. Find the maximum rectangle that can be inscribed in a circle of 
radius a. Let 2 x = base and 2 y = altitude; then 
area A = 4: xy =4:X Va^ — x^. 




Take / (x) = x^ (a^ - x"^) = a^x^ - x^ [by Art. 87]; 

/' (x) =2aH-4:X^ = 2x (a^ -2x^) = 0; .-. x = 0, a/V2; 
J" {x) =2a^-12x^; f" (0) = 2 a^; /. / (0) = is min. 
f"ia/V2) = 2o? - 6a2 = -4a2; .-. /(a/V2) = aV4. 
/. A = 4 V^ = 2 a2 

is the area of the maximum rectangle, which is a square. 
Note. — By Geometry without the Calculus method: 

A =2ay = 2aVa^ - x'-]j^o = 2a2, 

since the radical quantity is evidently greatest when x = 0. 




2. The strength of a beam of rectangular cross section varies as the 
breadth 6 and as d^, the square of the depth. Find the dimensions of 
the section of the strongest beam that can be cut from a cylindrical 



128 



DIFFERENTIAL CALCULUS 



log whose diameter is 2 a. Strength oc bd^; .'. strength = kbd^, where 
fc is a constant; let 

fih)=b (4a2 - 62) = 4,a% - b^; 

V-3(2a). 
2a 2.. „, 16a3 



/'(6) =4a2- 362=0; 



6 = -^, d 

V3 



/ (6) = ^ (4 a2 - ^) = -r^ . = (4 a^) = 



Hence, the rectangle may be laid off on the end of the log by drawing 
a diameter and dividing it into three equal parts; from the points of 
division drawing perpendiculars in opposite directions to the circum- 
ference and joining the points of intersection with the ends of the 
diameter, as in the figure. The strength of the beam is about 0.65 of 
that of the log, but it is the strongest beam of rectangular section. 

3. The stiffness of a rectangular beam varies as the breadth 6 and as 
d^, the cube of the depth. Find the dimensions of the stiff est beam that 
can be cut from the log. 




Stiffness oc 6^^; .*. stiffness = hbd^ {k constant); let stiffness 
= 6 (4 a2 - 62)3. Take 

/(6) = 4a%" - 6«; f (6) = f (a26-' - 6§) - 0; 
/. 62 = a2 or 6 = a; :. d = (4a2 - a^)^ = a Vs. 

To draw the rectangle, lay off from ends of a diameter chords at angles 
of 30° with diameter and join ends of chords with ends of diameter. 

4. A square piece of pasteboard is to be made into a box by cutting 
out a square at each corner. Find the side of the square cut out, so 
that the remainder of the sheet will form a vessel of maximum capacity. 
Let a be side of square sheet and x side of square cut out; then 



EXERCISE XI 



129 



f{x) =x{a- 2xY. 
fix) = -4:x{a-2x)^{(i -2xy = (a - 6 x) (a - 2 x) = 0; 

/. a = 1/6 a, 1/2 a. 

/ (1/6 a) = 1/6 a (a - 1/3 a)^ = 2/27 a^, maximum capacity. 
/ (1/2 a) = 1/2 a (a — a) =0, minimmn. 









j: 


a-2x 


X 







5. A rectangular sheet of tin 15" X 8" has a square cut out at each 
corner. Find the side of the square so that the remainder of the sheet 
may form a box of maximum contents. 

Ans. \\". 

6. A channel rectangular in section, carrying a given volume of 
water, is to be so proportioned as to have a mini- 
mum wetted perimeter. Find the proportions of _^^_ 
the channel. ; 

Let X be the width of the bottom, and y the ^ 
height of the water surface. Since the given ^^ 
volume is proportional to the cross section, 



xy = V, where V is constant. 

2V 

p ^ X + 2y = X -\ , from (1) ; 

X 



m 



that is, 



^ = 1-^ = 

dx ^ x^ ^' 



2V 



X = VJV = \/2 



2xy, or x = 2 y. 



^y\ 



To show that this makes p a minimum; note that for a:^ < 2 7, -^ is 

negative, and for x^> 2V, -^ is positive, therefore for x^ = 2 7, or 
a? = 2 2/, p is a minimum. 



130 DIFFERENTIAL CALCULUS 

7. Find the dimensions of a conical tent that for a given volume will 
require the least cloth. 

V = ^ Trr%; :. h = S V/irr^, where r is radius of base and h altitude. (1) 

^ = Trr Vr2 + /i2 = Trr {r^ + 9 FV^V)^ = (ttV + 9 FVr^)^ 
•S denoting lateral surface; let 

/ (r) = TT^r^ + 9 7Vr2 (by Art. 87), 

/ (0 = 47r2r3 - ^;^ =0, .-. r = — y — ; 
f" (r) = 127r2r2 + 54 F2/^^ positive for any r; 

hence r = —r- V/ — makes ^ a minimum. 

From (1), A = V — = r V2; and slant height = r Vs. 



8. From a given circular sheet of tin, find the sector to be cut out so 
that the remainder may form a conical vessel of maximum capacity. 

Ans. Angle of sector = (l- V|) 2 7r = 66° 14'. 

9. The work of propelling a steamer through the water varies as the 
cube of her speed; show that her most economical rate per hour against 
a current running n miles per hour is 3 n/2 miles per hour. 

Let V = speed of the steamer in miles per hour; 

then cv^ = work per hour, c being a constant ; 

and V — n = the actual distance advanced per hour. 

Hence, cv^/v — n = the work per mile of actual advance. 

Find the most economical speed against a current of 4 miles per hour. 

10. The cost of fuel consumed by a steamer varies as the cube of her 
speed, and is $25.00 per hour when the speed is 10 miles per hour. The 
other expenses are $100 per hour. Find the most economical speed. 
Let C = cost per hour for fuel at speed of v miles per hour; 



EXERCISE XI 131 

then C : $25 = «;3 : (lO)^; /. C = $25 y3/(iO)3; 

f(v)= -TTTrrr- • - i ; where a is distance and - is hours. 

(10)3 V V V 

y3 = 2000, or y = ^^2000 = 12.6 miles per hr. 
/ (12.6) = $12 cZ (approximately); 

hence cost for one hour about $150; cost for running 10 miles at 12.6 
miles per hour about $120, while the cost for running the 10 miles at 
10 miles per hour is $125, and at 15 miles per hour the cost for running 
10 miles is about $123. 

11. The amount of fuel consumed by a steamer varies as the cube of 
her speed. When her speed is 15 miles per hour she burns 4| tons of 
coal per hour at $4.00 per ton. The other expenses are $12.00 per hour. 
Find her most economical speed and the minimum cost of a voyage of 
2080 miles. Ans. 10.4 mi. per hr.; $3600. 

12. A vessel is anchored 3 miles off shore. Opposite a point 5 miles 
farther along the shore, another vessel is anchored 9 miles from the 
shore. A boat from the first vessel is to land a passenger on the shore 
and then proceed to the other vessel. Find the shortest course of the 
boat. 

Let hi be the distance the boat goes from first vessel to shore and h2 
the distance from the shore to the other vessel; then 

/ (x) =hi+h = (32 + x-^y^ + [92 + (5 - x)]^ 

f (x) = - 5 — X ^ 

^ ^ (9 + a;2)^ (81 + (5 - a;)2)i 

whence re = ± |; h + hi = --i- -\- -^-= 13 miles. 

13. Find the number of equal parts into which a given number n 
must be divided that their continued product may be a maximum. 
Let 

m=(^y; f(.) = g)(log^l)=0; 

. . log — = 1 ; - = e, and x = — , 
"^ X ' X ' e 

hence the number of parts is n/e, and each part is e. 



132 DIFFERENTIAL CALCULUS 

DETERMINATION OF POINTS OF INFLEXION. 

1. Examine y = x^ — S x^ — 9 x -\- 9; for points of inflexion. 

^ = 3x2-6x-9, 
ax 

^ = 6a; - 6 = 6 (x - 1) = 0, .-. x = l, 
ax^ 

is abscissa of an inflexion point. The point is (1, —2), to the right of 
which the curve is concave upward. 

2. Examine a;^ — 3 hx^ + a^y = 0, for points of inflexion. 

Ans. (b, 2 ¥/a^) is a point of inflexion, or of maximum slope, 
to the right of which the curve is concave downward. 

3. Examine y = c sin a; for points of inflexion. 

Ans. (0, 0), (±7r, 0), (±2 7r, 0) . . . . 

4. Examine the Witch of Agnesi, y = ^ , for inflexion points. 

Ans. (±2 a/Vs, 3 a/2); concave downward between these points, 
concave upwards outside of them. 

Find the points of inflexion of the following curves: 

5. (x/a)2 + (y/b)i = 1. Ans. x = ±a/V2. 

6. y = {x^ + x) e~^. ' Ans. a; = and a; = 3. 

1. y = e-''-e-^. 4^. 2(Mii^log6). 

a — h 
8. y = x^a^ + a:2. 

Ans. (0, 0), (a Vd, 3 a Vf), (-a Vs, -3 a Vf). 



CHAPTER VI. 



CURVATURE. EVOLUTES. 

88. Curvature. — The flexion (Art. 13), 6 =^ = ^, 

being the rate of change of the tangent of the angle made 
with the X-axis by the tangent to a curve, is one measure of 
the bending of the curve at the point of tangency. This 
measure, however, is dependent upon the position of the 
axes and would change if the axes were rotated. 

There is a measure of the bending called the curvature, 
which does not depend upon the choice of the axes, as it is 
expressed in terms that are the same after the axes are 
rotated, or even before any axes are drawn. The curvature 

is denoted by K = ~^, the rate of change of the angle of 

inclination = tan"^ m, with 
respect to the length of arc s. 

Thus, let P and Pi be two 
points on a plane curve, and 
</) + A0 the angles which the 
tangents at P and Pi make with 
the X-axis, s the arc AP meas- 
ured from some fixed point A 
on the curve up to P, and As 

the arc PPi. The angle is in radians, and A0 is evidently 
the angle between the two tangents. 

The angle A0 is the total curvature of the arc As, as it is a 
measure of the deviation from a straight line of that portion 
of the curve between the points P and Pi. The sharper the 

133 




134 DIFFERENTIAL CALCULUS 

bending of the curve between the two points the greater is 
A(/) for equal values of As. 

The average curvature of the arc As is defined as -r-^, and 

is, therefore, the average change per unit length of arc in 
the inclination of the tangent hne. 

The limit of -r— , when Pi approaches P as its limiting 

position, is called the curvature of the curve at P; that is, 

the curvature at a point on a curve is K = lim -z— = -^' 

As=o As ds 

Otherwise, by rates, the curvature of any curve, as APPi 
at any point, as P, is the s-rate at which the curve bends at 
P, or the s-rate at which the tangent revolves, where s de- 
notes the length of the variable arc AP. If <^ denotes (in 
radians) the variable angle XTP as P moves along the curve 
APPi, then, evidently, the curvature of APPi at P equals 

the s-rate of <^ ; that is, X = -p- • 

89. Curvature of a Circle. — For a circle of radius R, 
As = RAcf) and therefore, 

^ _ 1 # _ 1 
As "P' ds~ R' 

since the ratio of the increments is con- 
stant ; that is, the average curvature of 
any arc of a circle is equal to the curva- 
ture at any point of that circle. In 
other words, a circle is a curve of constant curvature and its 
curvature is equal to the reciprocal of its radius; that is, 
the curvature of a circle equals 1/P radians to a unit of 
arc. 

For example, if P = 2, the circle bends uniformly at the 
rate of J radian to a unit of arc. 




CIRCLE, RADIUS, AND CENTER OF CURVATURE 135 

If i^ = I, the curvature of the circle is 2 radians per unit 
of arc. 

If E = 1, for the circle of unit radius the curvature is 
evidently a radian per unit of arc. 

90. Circle, Radius, and Center of Curvature. — The 
curvature of any curve except the circle varies from one 
point to another. A circle tangent to a curve and having 
the same curvature as the curve at the point of contact, 
therefore, having a radius equal to the reciprocal of the 
curvature at that point, is called the circle of curvature at 
that point; its radius is called the radius of curvature; and 
its center, the center of curvature. 

If R denotes the radius of the circle of curvature at any 

point of a curve, then, since the curvature of the curve is j- 
and equals the curvature of the circle, it follows that, 

If at P (Art. 72, figure) the direction of the path of (x, y) 
became constant, {x, y) would trace the tangent at P, and ds 
might be represented by a length on the tangent ; while if at 
P the change of direction of the path became uniform with 
respect to s, (x, y) would trace the circle of curvature at P, 
and ds would represent an arc of the circle, since it equals 
R d(t), where d(t) = ^4> would be the constant change of angle 
at center of the circle of curvature. 

Thus it is that the curvature is uniform when, as the 
moving point passes over equal arcs, the tangent turns 
through equal angles; or conversely; and, as this is the case 
with the circle only, it is the only curve of uniform curvature. 

For any curve the measure of the curvature at a point is 
the limit of the ratio between the angle described by the 
tangent and the arc described by the point of contact, as 

that arc approaches zero; and this limit y- equals the 



136 



DIFFERENTIAL CALCULUS 



reciprocal of the radius of curvature at the point; hence, 

ds 
the radius of curvature is -r- and equals the reciprocal of the 

curvature. The figure shows the circle of curvature for the 
point P {x, y) of the ellipse; C is the center of curvature, 
and CP the radius of curvature. It is to be noticed that the 



B 


\^^ 


y^^ 


"''r^ 


\ 1 " 


J(y\\ 


. ^ 





circle of curvature crosses the ellipse at P, and this must be 
so; for at P the circle and elUpse have the same curvature, 
but towards A the curvature of the elUpse increases, while 
that of the circle remains the same, being constant. Hence 
on the side of P towards the vertex A the circle is outside of 
the ellipse. From P towards B the curvature of the ellipse 
decreases, and, therefore, on the side of P towards the vertex 
B the circle is inside of the ellipse. 

So, in general, the circle of curvature crosses the curve at the 
point of contact. 

The only exceptions to this rule are at points of maximum 
and minimum curvature, as the vertices of the ellipse. From 
A along the curve in either direction, the curvature of the 
ellipse decreases; hence the circle of curvature at A Ues 
entirely within the ellipse. From B the curvature of the 
ellipse increases in each direction and so the circle of curva- 
ture at B lies entirely without the eUipse. 



RADIUS OF CURVATURE IN COORDINATES 137 

91. Radius of Ctirvature in Rectangular Coordinates. — 

Since = tan"^ m, and since ds^ = dx^ + dy^; 

d(b = :; — I — r and ds = V 1 + m^ dx, where m = ^ : 
1 + m^ ax 

hence the radius of curvature 

Tf _ds _ Vl + m^ dx _ (1 + m^)i 
d<f} dm dm 



1 +m^ dx 

(l+m^)t _ L^+(dj 
6 d'^y 



(1) 



•dx2 



R will be positive or negative according as -r^ is positive or 

negative, if is always taken as the acute angle which the 
tangent makes with the x-axis; for then, whether is posi- 
tive or negative, 1 "I" ( j^J f = sec^0 will be positive and 

d^v 
the sign of R will be that of -7^. Hence the sign of R will be 

plus or minus according as the curve at the point is concave 

sec^ d) 
upward or downward. R may be in the form — 7 — • 

If the reciprocals of the members of (1) are taken, then 

^ = S = S/[^ + 07' ^hi«i'-'"*y be in the form 
h cos^ <^. 

If -j-^ is zero at any point of a curve, then K = -^is zero 

and R is infinite. Thus at a point of inflexion R is infinite. 
It may be noted that as a curve approaches being a straight 
line, its curvature approaches zero and its radius of curvature 
becomes infinite, that is, it increases in length without limit. 
So a straight line is the line that the arc of a circle of curva- 



138 DIFFERENTIAL CALCULUS 

ture approaches as the radius of the circle increases without 
limit. On the contrary as the radius of curvature at a point 
approaches zero, the curvature at the point becomes infinite 
and the curve will approach a mere point, since the circle of 
curvature will diminish with zero as a limit for its radius. 
92. Approximate Formula for Radius of Curvature. — 

Smce K 



Mi)T 



it is seen that the flexion when multiplied by the factor 
1/(1 + m^)^ gives a measure of the bending of a curve 
independent of the position of the axes. 

The flexion is the rate of change of the tangent of the inclina- 
tion of the curve at a point with respect to the abscissa, while 
the curvature is the rate of change of the inclination of the 
curve at a point with respect to the arc, where the inclination 
of the curve is that of the tangent line at the "^oint. 

However, when the curve deviates but slightly from a 
horizontal straight line, the curvature is approximately the 

same as the flexion, since the slope m = ~ being small, l-^) 

is very small compared with 1, and therefore the formula 
becomes approximately 

This approximation for the curvature is used to advantage 
in the flexure of beams and columns. The approximate 
formula for the radius of curvature is consequently 



EXERCISE XII 139 

EXERCISE Xn. 

1. To find R and the curvature of the ellipse -^ + r^ = 1. 

^ = _ ^ ^ = _ _^. 
dx a^y' dx^ a^y^ 

Substituting these values in (1), Art. 91, gives 

V^aY) \ ¥ J ^i5i ' 

^ d<f>_ 1 ^ a'¥ 

ds R {aY + hH'')^' 

The maximum curvature is a/¥, at A (a, 0), where R = ¥/a is a 
minimum, and the minimum curvature is b/a% at B (0, h), where 
H = a^/b is a maximum. (See Art. 90, figure; Art. 97, figure.) 

2. To find R and the curvature of the parabola y^ = 4 px. 

dy ^2p ^ ^ _ ^P\ 
dx y ' da;2 y^ 

Substituting these values in (1) of Art. 91 gives 

±P)} 



f y' + iy^ Yf y'\_ 2 (x + 



ds R 2{x + p)^' 

At the vertex (0, 0), R = 2 p, the minimum radius; and the maxi- 
mum curvature is (1/2 p) radian to a unit of arc. (See Example 1, 
Art. 97.) 

Since -r^ = j- is negative for positive values of y and posi- 
tive for negative values, the curve is concave downward at points 
whose ordinates are positive, and concave upward at points whose 
ordinates are negative. The sign of R may be neglected, since the 

sign of -T-| will indicate whether the curve is concave upward or down- 
ward at any point. 

3. To find R of the cycloid x = a vers"^ {y/a) ^ ^2 ay — y^. 

dy _ V2 ay — y^ d'^y _ a ^ 
dx y ' dx^ ?/2 



140 DIFFERENTIAL CALCULUS 

Substituting these values in (1) of Art. 75 gives 

At the highest point, y = 2 a, and, therefore, R = 4: a, the maximum of 
R. At the vertex (0, 0), R = 0, and also at other points where y is 
zero; therefore, R being zero, at those points, which are cusps, the 
curvature is infinite. (See Example 3, Art. 97, figure.) 
Find R and the curvature of each of the following curves. 

4. The equilateral hyperbola 2xy = o?. E = {x^ + 2/^)^/^^. 
6. The cubical parabola y^ = o?x. 

6. The logarithmic curve y = 6*. 

7. The catenary i/ = ^ (e^/«+ e"^/'^). 

8. The hypocycloid x' + 2/' = a^. 

9. The curve x^ -\- y^ = aK 

93. Radius of Curvature in Polar Coordinates. — From 

(4) and (10) of Art. 67, 

^ = 6 + ^, ^ = tan-i^; 

•*• dd~ '^ dd' dd p2 _|_ (^dp/doy 

d4> ^ p^-^2 {dp/dey - p . (IV/c^6>^ , 

•• de p' + {dp/dey 

. ^ ds/dl ^ [p^ + (^p/(i^)^]^ ,. 77 .ON 

•• ^ dct>/dd p^ + 2(dp/ddy-p'd'p/dd''' ^^^-^'^^^ 

Corollary. ■. — Since R = cc at a point of inflexion, 

p^ + 2{dp/dey-p.^,= o 

is a necessary condition for such a point. 



K = 


d4> 
^ ds ' 




6aV 


(9 


y' + a')^ 


dcj) 




my 




ds 


{m 


' + y 


.)i 


d<l> 
ds '' 


a 






R = 


- 3 {axy)K 




K = 


d(i> 




a^ 


2( 


x + y)^ 



RADIUS OF CURVATURE IN POLAR COORDINATES 141 



Example. — To find R for the curve p = sin ^. Here 
dp 
dd 






R = 



(p^ + cosH)i 



(sin2^ + cos2 6i)^ 



p2+2cos2^-p(-sin6') sin2 0+2cos2^+sin2^ 2 

This curve p = sin 6, a circle with ^ 

unit diameter, in connection with the 
formula for polar curves, tan \l/ = 

;, furnishes a derivation of the 



dp/dd 

d (sin 6). Since for circle 




\l^ = d, tan d 



dp/dd 



dp 



p dd sin 



tan 6 tan 

that is, d (sin 6) = cos 6 dd. 
Also from figure : 

OP p 



dd = cos ddd; 



tSLlid = 



OA cos (9' 



and since 



tan^ 

dp 
dd 



dp/dd' 



= cos d = subnormal OA ; 



so again, d (sin d) = cos d dd. 

This curve serves as an illustration of maxima and minima 
in polar coordinates. Thus, p = sin will be a maximum 



or a minimum when -^ = cos = 0, 



when ^ = - or 



-^ ; and since -^^ = — sin d, is negative when ^ = o > 

. TT ^ . . , ., d'^p • /. • 

p = sm - = 1 IS a maximum, while -^ = — sm is posi- 



142 



DIFFERENTIAL CALCULUS 

— 1 is a minimum. 



trv€ when 6 = -x- , .*. p = sm 



2 ' • • ^ -' 2 
As the denominator of the fractional value of E is 2 for any 
value of d, there is no inflexion point, R not being infinite 
at any point. 

EXERCISE Xni. 

Find R in each of the following curves : 

1. The Cardioid p = a (1 — cos0). 

2. The Lemniscate p^ = a^ cos 2 0. 
Where is an inflexion point ? 

3. The Spiral of Archimedes p = ad. 



R = 2V2ap/3. 



+ 2 



4. The Logarithmic Spiral p = a^. 



R = a 

R = pVl + (loga)2. 



94. Coordinates of Center of Curvature. — Let P (x, y) 
be any point on the curve ab, and C {a, (3) the corresponding 




to 



center of curvature. Then PC is R and is perpendicular 
the tangent PT. Hence 

Z BCP = Z XTP = 0, 

OA = OM - BP, AC = MP-\- BC; 

a = x — Rsm(i), ^ = y -\- Rcos(f); (1) 

T^dy ^ , -^dx 



that is, 
or 



(2) 



PROPERTIES OF THE INVOLUTE AND EVOLUTE 143 



Substituting in (2) . the values of R and ds, gives 



a = X 



dx" 



1 + 



^ = y + 



1^1 

\dx/ 



d^ 
dx' 



(3) 



95. Evolutes and Involutes. — Every point of a curve, as 

in, has a corresponding center of curvature. As the point 
(x, y) moves along the curve in, by equation (3) above, the 




point {a, ^) will trace another curve, as ev. The curve ev, 
which is the locus of the centers of curvature of in, is called 
the evolute of in. 

To express the inverse relation, in is called an involute of 
ev. The figure shows an arc of an involute of a circle. 

96. Properties of the Involute and Evolute. — I. Since 

dx/ds = cos cf), dy/ds = sin 0, and ds = R dcj), 

dx = cos (l)ds = R cos dcj), (1) 

and dy = sin (jtds = R sin </> dcf). (2) 

Differentiating equations (1) of Art. 94, and using the 
relations given in (1) and (2), there results 

da = dx — R cos <f> d(f) — sin 4> dR = — sin dR, (3) 
d^ = dy — R sin <l)d<f> -\- cos 4)dR = cos </> dR. (4) 



144 DIFFERENTIAL CALCULUS 

Dividing (4) by (3) gives 

d^/da = — cot </) = — dx/dy. 

That is, the normal to the involute at {x, y),as P (Art. 95, figure), 
is tangent to the evolute at the corresponding point (a, /3), as C. 
II. Squaring and adding (3) and (4) gives 
da^ + d/32 = dRK 

Let s denote the length of an arc of the evolute; then, 

da^ + d^^ = ds\ 

Hence, rfs = :±:dR; .*. As = zb Ai?. 

That is, the difference between two radii of curvature, as 
C3P3 and CiPi (Art. 95, figure), is equal to the corresponding 
arc of the evolute, C1C3. 

These two properties show that the involute in can be 
described by a point in a string unwound from the evolute 
o). From this property the evolute receives its name. 

It may be noted that a curve has only one evolute, but 
an unlimited number of involutes, as each point on the string 
which is unwound would describe an involute. 

97. To Find the Equation of the Evolute of a Given 
Curve. — Differentiating the equation of the given curve and 
using equations (3) of Art. 94, a and jS will be expressed in 
terms of x and y. These two equations and that of the given 
curve furnish three equations between a, j8, x and y. Ehmi- 
nating x and y from these equations, a relation between a 
and /3 is obtained, and this relation is the equation of the 
evolute of the given curve, which would itself be an involute of 
the curve found. 

Examples. — Find the equation of the evolute of the 
following curves: 

1. The parabola y^ = Apx. (1) 

dy _2p d?y _ _ 4p^ 
dx y ' dx^ y^ 



EQUATION OF THE EVOLUTE 
Substituting these values in (3) of Art. 94 gives 

:. x={oi-2 p)/3, 7/ = - v'4^2. 
Substituting these values of x and ?/ in (1) gives 
i32 = 4 (a- 2 p)V27p, 

as the equation of the evolute of 2/^ = 4 yx. 

The locus of (2) is the semi- 
cubical parabola. Thus, if 
iOn is the locus of (1), F 
being the focus, then eAv is 
the locus of (2) , where OA = 
2 7? = 2 OF is the minimum 
radius of curvature at 0, the 
point of maximum curvature 
on the parabola. (Example 
2, Exercise XII.) 

2. The ellipse 
ay + 6V = o'hK (1) 

d^ _ _b'^x d}y _ _ h^ 
dx a^y^ dx^ a^y^ 

Substituting these values in 
(3) of Art. 94 gives 

(a^ — 6^) x 



145 



/3 = 



(a^ - 6^) y^ 



¥ 



W -hy ' ^ w - hy 



(2) 




Hence, the equation of the evolute of the eUipse aV + Ifx^ = 
a^b^ is 

(aa)^ + (h^ = (a' - 6^)3. 

The evolute is C1C2C1C2. Ci is center of curvature for 



146 



DIFFERENTIAL CALCULUS 



A; C for P; C2 for B; Ci' for A'] C2' for B'. In the figure 
shown a = 2h; when a = h V2, then the center of curvature 
for B is Sit B' and for 5' at 5. When a <h V2, the centers 

for B and B' are within 
the elHpse. The points Ci, 
C2, Ci, and C2 are cusps. 
The length of the evolute 
is evidently four times the 
difference between R at 
B {a, h), and R Sit A (a, 0); 
that is, (1, Exercise XII), 
4 {ayb - by a) = 4 (a^ - h^) 
/ah. 

Corollary. — For circle, 
since a = b, the evolute is 
a point, the center of the 
circle. 

Y\ 





The involute of the circle is 
given by the equations, 

X = a(cos^ + 0sin0), 
y = a{smd — 0cos^). 

AP is the arc of an involute of the circle. 

3. The cycloid x = a vers~^ (y/ct) ^ ^2 ay — y^. 

dy _ V2 ay — y'^ (Py _ a 
dx y ' dx^ y^ 

Substituting these values in (3) of Art. 94 : 

2/=- ft x = a = 2 V- 2ai8-i32; 

.-. a = avers-i (-/3/a) db V -2al3- 13^. 



(1) 



The locus of (1) is another cycloid equal to the given one, the 



EQUATION OF THE EVOLUTE 



147 



highest point being at the origin; that is, the evolute of a 
cycloid is an equal cycloid. Thus, the evolute of the arc OPi 
is the arc OCi, which equals Pix; and the evolute of Pix is 
Cix, which equals OPi. 
Since R = 2 V2^ (3, Exercise XII), CiPi = 4 a. Then 
OPiX = 2 . OCi = 2 . CiPi = 8a. 

Hence, the length of one branch of the cycloid is eight times 
the radius of the generating circle. (See Example 3, Art. 
142.) 

P. 




If the figure shown be inverted, the principle of the cy- 
cloidal pendulum may be perceived. A weight, suspended 
from Ci by a flexible cord, may be made to oscillate in the 
arc OPiX, by means of some surface shaped hke the arcs 
CiO and CiX causing a continuous change in the length of the 
cord as it comes in contact with the surface. The cycloidal 
pendulum is isochronal, as the time of an oscillation is 
independent of the length of the arc. (See Art. 237.) 

4. The hyperbola 6V — a'^y^ = a^"^. 

(aa)^ - (6/3)t = (a' + h')\ 

5. The equilateral hyperbola 2xy = d^. 

(a + ©^- (a-^)3 = 2al 

6. Find the length of an arc of the evolute of the parabola 
^2 _ 4 p^ in terms of the abscissas of its extremities. 



148 DIFFERENTIAL CALCULUS 

Arc AC = CP-AO = ^(^ + ?^)' -2p (Example 1, figure) 

7. Show that in the catenary y = a/2 \e^ + e °/, 

a = X — y/a Vy^ — a^, ^ = 2y. 

8. Find the equation of the evolute of the hypocycloid 
X3 -\-y3 = a^, 

(Q: + i8)3 + (a - ^)i = 2ai 



CHAPTER VII. 

CHANGE OF THE INDEPENDENT VARIABLE. 
FUNCTIONS OF TWO OR MORE VARIABLES. 

98. Different Forms of Successive Derivatives. — As 
given in Arts. 67, 68, where x is independent dx may be 
taken as having always the same value and is accordingly 
treated as a constant; hence, 

d dy _ d^y d_ d^ dy _ d^y 
dx dx dx^' dx dx dx dx^' 

When neither x nor y is independent, -^^ is a fraction with 

both numerator and denominator variable, and d dx = dH, 
etc., hence, 

d dy _dx d'^y — dy dH , . 

dx dx dx^ ' 

d d dy _ dx^ d^y — dx dy d^x — 3dx d^x d^y -\-Zdy (d'^xY , . 
dx dx dx dx^ > ^ J 



When y is independent, d'^y = 0, d^y = 0, . . . ; hence 
d dy dy dH 



dx dx dx^ ' 

d d dy _Sdy (d'^xY — dx dy d^x 
dx dx dx dx^ ^ 



(10 

(20 



99. Change of the Independent Variable. — In some ap- 
plications of the Calculus it is necessary to make a differen- 
tial equation depend on a new independent variable in place 
of the one originally selected; that is, there is need to change 
the independent variable. 

149 



150 DIFFERENTIAL CALCULUS 

When X = cl){z) and it is desired to change the independent 

variable from x to z; ior -r-^, -p^, . . , , respectively, the 

second members of (1), (2), . . . , above, are substituted; 
and in the resulting equation, for x, dx, dH, . . . , their 
values gotten from the equation x = ^{z) are substituted. 

Example 1. — Given y d'^y + dy^ + dx^ = 0, in which x is 
independent, to find the transformed equation in which 
neither x nor y is independent; also the one in which y is 
independent. 

Dividing both members by dx"^, substituting for j^ the 

second member of (1) Art. 98, and multiplying both mem- 
bers by dx^, gives 

y {d^y dx — dH dy) + {dy"^ dx + dx^) = 0, 

in which neither x nor y is independent. 

Putting d'^y = 0, and dividing by —dy^, gives 

dH _ dx^ ^^ _ n 

in which the position of dy indicates that y is independent. 

Example 2. — To change the independent variable from 
a; to in 

^ [l + (dy/dxY]\ 
d^ 
dx" 

given X = p(ios6,y = p sin 6, p being a function of 0. 
From the data, 

dy — sine dp -{- p cos 6 d6, 

dx — cos 6 dp — p sin d dd, 
d^y = sin dd^p + 2 cos ddddp- p sin 6 dd^, 
and 

d^x — cos dd^p — 2 sin ddddp — p cos 6 dd^. 



EXERCISE XIV 151 

Substituting these values in value of R and simplifying, gives 
U = [P^ + (^f /^^)^i;^ , the value of R in Art. 93. 



P^ + 2 



(dpV_ (Pp 
\dd/ ^ dd' 



To change the independent variable from x to y; for 
d^y/dx^, d^y/dx^, . . . , respectively, the second members of 
(10, (20, • • • , above, are substituted; or in the general 
result, as in example 1, make d'^y = 0, d^y = 0, etc. 

Example 3. — Change the independent variable from x to 
2/ in 

\dx'^/ dx dx^ dx^ \dx] 

d^v d^xi 

Substituting for -t\ and -7-|, respectively, the second mem- 
bers of (10 and (20 gives after reduction 

d^x dH _ ^ 
d^'^dy^~ ' 

in which the position of dy shows that y is independent. 



EXERCISE XIV. 

1. Given x = cos d, change the independent variable from a; to in 

2. Given x = -, change the independent variable from x to z in 

3. Given x^ = 4 z, change the independent variable from x to 2 in 

dx^ X dx ^ dz^ dz ^ 

4. Given x = cos z, change the independent variable from x to 2 in 



152 DIFFERENTIAL CALCULUS 

5. Change the independent variable from x to ?/ in 

6. Given z = , . , , to find the transformed equation when 

X = pQosd,y = p sin 6, and p is independent. Ans. z = p-^r' 

dp 

100. Function of Several Variables. — A function may 
depend upon two or more variables having no mutual rela- 
tion, that is, independent of each other. Thus the volume 
of a gas depends upon the temperature and also upon the 
pressure to which it is subjected, and the temperature and 
the pressure may vary independently. 

A variable ^ is a function of the independent variables x, 
y, . . . when for each set of values of these variables there 
is determined a definite value or values of z. 

A function of two variables 

^ = f(^,y)y where x and y are independent, 

is represented geometrically by a surface, plane or curved 
according to the form of the functional relation ; and to each 
pair of values of (x, y) there corresponds a point on this 
surface. When x and y vary, the point takes another posi- 
tion, and it will take the new position either by x and y 
varying simultaneously or by one remaining constant while 
the other changes. 

101. Partial Differentials. — A partial differential of a 
function of two or more variables is the differential when 
only one of the variables is supposed to change. Let z = 
f (x, y) be the surface shown in the figure, and P{x, y, z) the 
moving point; then if y is constant while x changes, P will 
move on the plane curve PA and dx may be represented by 
PM or P'M'; on the other hand, if x is constant while y 
changes, P will move on the plane curve PB and dy may be 
represented by PN or P'N'. The differential of ;s as a 



PARTIAL DIFFERENTIALS 



153 



function of x, y being regarded as a constant, is denoted by 
dxZ) and the differential of z when y alone is variable is 
denoted by dyZ. These differentials are the partial differ- 
entials of z with respect to x and y, respectively. 




Note. — The partial dxZ may in the figure be represented 
by the distance on the ordinate from the point M to the 
tangent TP, and so too, the partial dyZ may be represented 
by the distance from N to the tangent T'P, both being 
negative in this case as z is decreasing. The A^js; and the 
Ay2: are the distances from the points M and N to the surface 
curves through P2. 

102. Partial Derivatives. — The partial derivatives of z 

with respect to x and y are denoted by t- and -j- , respectively, 

and they may be represented by the equivalent notation, 
fJix, y) and //(a;, y). 

In the figure of Art. 101, consider P as the intersection of 
the curves CPA and C'PB, cut from the surface by the 
planes y = b and x = a, respectively; then the slope of the 



154 DIFFERENTIAL CALCULUS 

curve CPA is given by the partial derivative -7- , and that of 

dz 

the curve C'PB by the partial derivative -7-; that is, the 

partial derivatives are the tangents of the inclination of the 

tangent lines at P to the axes of X and F, respectively. The 

values of the slopes for some definite point P on the surface 

dz dz 

are gotten by substituting in the expressions for — and -j- , 

respectively, the corresponding values of x and y. Thus in 
this case (a, h), or P\ being the projection of P on the xy- 
plane, a is substituted for x and b for y. 

103. Tangent Plane. Angles with Coordinate Planes. 
— In the figure of Art. 101, let P be the point (xi, yi, Zi); 
PT, the tangent to CPA in the plane y = yi] and PT' , the 
tangent to C'PB in the plane x = Xi. 

The equations of PT are 

z-zi = \^A (^ - ^1)' y = Vh (1) 

and of Pr, 

z-zi = \^\ (y - yi), X = xi. (2) 

The plane tangent to the surface at P has for its equation, 

since it is determined by the two intersecting tangents, is 
of the first degree with respect to x, y, z, and is satisfied by 
(1) and (2). 

The equations of the normal through P are those of a line 
through {xi, yi, Zi) perpendicular to (3). Its equations are 

The angles made by the tangent plane with the coordinate 'planes 
are equal to the inclinations of the normal to the axes. 



Zi 



TANGENT PLANE 155 

The direction cosines of the hne perpendicular to (3) are 

proportional to [|]_,[|],-1. 

Hence, if a, (3, y, are the inchnations of the normal to OX, 
OY, OZ, respectively, 

Also cos^ a + cos^ jS + cos^ 7 = 1- (6) 

From (5) and (6), in general, at any point (x, y, z), 

— (11/ ■+(!)■+ (I)' 

— /' + (e)'+(S)" 

For the inclination of the tangent plane to XY, from (7), 

tan^. = (|J + (|J. . (9) 

From (9), calling the tangent of the angle made by the 
tangent plane with the plane XY the slope, 



"■"-=\W^- 



Example. — Find the equations of the tangent plane and 
normal, to the sphere x'^ -{- y^ -\- z^ = a^, at (xi, yi, Zi), 

dz _ _x §^ — _y. 

'dx~ z' dy ~ z' 

. r^l = _ E.1 r^l = -^1. 
Idxji Zi' [dyji Zi' 



156 DIFFERENTIAL CALCULUS 

Substituting in (3), 

z-z,= --{x-xi)-f-{y- yi), 

Zi Zi 

xxi + yyi + zzi = Xi^ + yi^ + Zi^ = a^. Ans, 
From (4) for the normal: 

(x -xi)^ = {y - yi) ~ = z- Ziy 

1 = — -1 = 1, —=^~ = -. Arts, 

Xi 2/1 ^1 ^1 2/1 ^1 

EXERCISE XV. 

1. Find the equation of the tangent plane and its slope, for the 
eUipsoid, x^ + 2y^ + Sz^ = 20, at (3, 2, -\-Zi). 

Ans. Sx + 4:y + 3z = 20; f. 

2. Find the equation of the tangent plane to the elliptic paraboloid, 
2 = 3 x2 + 2 ?/2, at the point (1, 2, 11). 

Ans. Qx + 8y — z = ll. 

3. Find the equations of the tangent plane and normal to the cone, 
3x2-2/2 + 222 = 0, at {xi,y,,z,). 

Ans. Sxx^ - yy, + 2zz^= 0; ^^ = ^-^^ = ^-^- 

o Xi —yi Zz\ 

Note. — The equations of the tangent plane and normal 
are illusory if formed for the origin. Every tangent plane 
to the cone goes through the origin and there is no definite 
normal at the origin. When at special points on a surface 
the three partial derivatives of the function with respect to 
each of the three variables are all zero, there is no definite 
tangent plane or normal at the point. Such points are 
called conical points, the vertex of a cone being the typical 
case. 

104. Total Differentials. — When z = f {x, y) is differ- 
entiated, both X and y varying, the total differential dz or 
df {x, y) is gotten. 

The derivations of the formulas for differentiation of al- 
gebraic, logarithmic and exponential functions, given in 



TOTAL DIFFERENTIALS 157 

Chapter II, hold when u, v, y, and z denote functions of two 
or more independent variables; hence the total differential 
of / (x, y) may be gotten by the principles estabUshed in 
those derivations. The total differential of a function of two 
or more variables is eqvxil to the sum of its partial differentials. 
liz = f(x, y), then 

dz = dxZ -\- dyZ = j-dx -\- J- dy; 
and if t; = f(x, y, z), then, 

dv = dxV -{- dyV -}- dzV = -r dx -{- -r dy -{- ^ dz, 
dx dy dz 

where the last form of the partial differentials is another 
convenient notation. In the figure of Art. 101, (Z;^ is repre- 
sented by the distance on the ordinate from D to the tan- 
gent plane at P and is there negative. As; being DP2, which 
is negative. 

The truth of this theorem has been illustrated geometrically 
in the derivations of d{uy) and d(xyz) in Arts. 28 and 29, and 
the theorem is readily established analytically. Thus, it has 
been found that all the terms of d{f(x, y)) are of the first 
degree in dx and dy; hence, ii z = f {x, y), 

dz = (l)(x, y) dx + 01 (x, y) dy, (1) 

where </)(x, y) and <^i(a;, y) denote, respectively, the sums of 
the coefficients of dx and dy in the several terms of dz. 
When a:; alone varies, (1) becomes 

dxZ = 4> (x, y) dx. (2) 

When y alone varies, (1) becomes 

dyZ = 01 (x, y) dy, (3) 

Hence, from (1), (2) and (3), 

dz = dxZ -\- dyZ = J- dx -^ J- dy. 



158 DIFFERENTIAL CALCULUS 

105. If. =/(.,,) = . |=-|g. (1) 

dz dz 

for -T- dx + J- dy = dz = d (c) = 0, (2) 

dv 
which solved for -^ gives (1). 

This formula for the derivative of an implicit function is 
useful in many cases. 

Example. — Given x^ — a^xy + ¥y^ = c = z,to find dy/dx. 

Here -r- = 4:X^ — a^y and -r- = — a^x + 2 ¥y; 

dy 4:X^ — o?y . i /-.x 

dx o?x - 2b^v -^ ^ ^ 



EXERCISE XVI. 

1. u'= xya^ + yy¥. 

du _2x du _2i 
dx a^ ' dy b^ 



X du .y du _ x^ y"^ 
2dx^2dy~ a^'^b^ 



2. u = b Xy''- + cx^ + ey^ 



^£ = bf + 2cx, ^^ = 2bxyi-Sey\ 



xy du , du 

6. « = log(e^ + .^), i + ^ = l. 

6. ■w = 6a;i/2 + cx"^ + 62/^. d^^ = {by"^ + 2 ca?) dx + (2 6x2/ + 3 ey^) dy. 

7. u = y^. du = y^ log ydx -\- xy^~^ dy. 

S. U == ?/sin X. du = ?/sin a; log y COS X da: + ^/sin^-^ sin X dy. 

By Art. 105, find dy/dx when: 

ax x^ — xy log x 



11. X log 2/ — 2/ log a; = 0. 



TOTAL DERIVATIVES 159 

dy y xlogy -y 



dx X y log X — X 



106. Total Derivatives. — li u = f {x, y, z), y = <l) {x), 
and z = (t)i{x),- u is directly a function of x and indirectly a 
function of x through y and z. The total differential, 

'^" = £^^ + S'^^ + £'^^ (by Art. 104) 
becomes by dividing by dx, 

du _ du .du dy du dz .^. 

dx dx dy dx dz dx' 

where -7- is the total derivative of i^ as a function of x. 
ax 

Corollary 1. — li u = f {y, z), y = 0(x), and z = <l)i{x), 

du _ du dy .du dz .^^ 

dx dy dx dz dx 

Corollary 2. — li u = f (y) and y = (f)(x), 

du _ du dy , . 

dx dy dx' 

du 
where -7- is the derivative of a function of a function, and 

(3) is the formula that is the subject of Remarks in Art. 19. 

Corollary 3. — If u = f {x, y, z) and x, y and z are inde- 
pendent of each other, they may be regarded as functions 
of time t; hence, the expression for the total differential du 
above becomes by dividing by dt, 

du _ du dx du dy du dz ,.. 

dt~dxdi^dy~di~^dzdi' ^ ^ 

where -^ is the total time-derivative or rate of change of u. 

Similarly, when z = f (x, y), x and y being functions of t, 

dz ^ dz^ dx dz_ dy . . 

Jt~ dx'dt^ dy~dt' ^^ 



160 DIFFERENTIAL CALCULUS 

If 2/ is a function of x, a.s y = <}>{x), putting x for t in (5), 
gives 

dx dx dy dx ^^ 

107. Illustrative Examples. — Example 1. — The edges 
of a right parallelepiped are 6, 8 and 10 feet. They are 
increasing at the rate of 0.02 foot per second, 0.03 foot per 
second and 0.04 foot per second, respectively. Show at 
what rate the volume is increasing. 

Let volume = u = xyz, then by (4), Art. 106: 
du dx , dy , dz 

Tt=y'Tt^'''Tt^''ydt 

= 80 X 0.02 + 60 X 0.03 + 48 X 0.04 

= 1.60 + 1.80 + 1.92 = 5.32 cubic feet per second. 

See Art. 29 where du{= dV) is shown geometrically by 
figure. 

Example 2. — Given the formula for gas, pV = KT, 
where p is pressure, V is volume, T is temperature, and K is 
a constant. Let K = 50, and let the volume and tempera- 
ture at a given time be Fo = 5 cu. ft. and To = 250°. The 
corresponding pressure is 

po = 5^-^p5? ^ 2500 lb. per sq. ft. 

If in this state the temperature is rising at the rate of 0.5 
degree per minute and the volume is increasing at the rate 
of 0.2 cu. ft. per minute, required the rate at which the 



pressure 


) is changing. 


Here p = 


T 
50^, 




whence 


dp 
dT 


50 


dp _ 
dV 


= -50 


T 
72 


Hence, i 


in the given state, 












dp- 
dT 


50 
5 


= 10, 





T=Tq 



ILLUSTRATIVE EXAMPLES 161 

, dpi 50 X 250 ^^^ 

V=Vo 

Given ^ = 0.5 and ^ = 0.2. 

at at 

Then 

by (5) Art. 106; that is, the pressure is decreasing at the 
rate of 95 lb. per sq. ft. per min. 

x^ iP" z^ 
Example 3. — A point on an elHpsoid ^ + f^ + in ~ 1? 

in the position x = S, y = — 4, moves so that x increases 
at the rate of two units per second, while y decreases at the 
rate of three units per second. Find the rate of change of z. 
Here 

dz^_ 7x_ a^ 7j 

dx _ dy _ 

Tt~^' Tt'"^' 

dz l^x 21 y 

di~ ~ "^ 



vi-S-l 25v/i-g- 



(by (5) 
Art. 106) 



-77 = * 7= units per sec, the rate of change of z. 

dt IsVll 

EXERCISE XVn. 

1. u = z^ -\- y^ + zy, z = sinx, y = e^; find -^ — 

Ans. -J- = 3 e^^ + e^ (sin x + cos x) + sin 2 x. 

2. u = V x2 + 2/2, y = mx-\- c. -j- = , 

dx Vx2 + (mx + c)2 



162 DIFFERENTIAL CALCULUS 



u - sin-1 {y - z), y = dx, z = 4: xK 


du 
dx 


w = tan-i^, aj2 _j_ 2/2 = r2. 

X 


du 
dx 

du 
dx 


u = log (x + y), y = Vx^-\- a\ 



VI -x^ 

1 



Vr2 
1 



6. With the same data as in illustrative Ex, 2, when the pressure of 
the gas is increasing at the rate of 40 lb. per sq. ft. per sec. and the 
temperature is falling at the rate of 1 degree per sec, find the rate of 
change of the volume. 

dV 
Ans. -J- = —0.1 cu. ft. per sec. 

7. A point on a elliptic paraboloid z = 2 x^ -\- 5 y"^, in the position 
x = —d,y = 1, moves so that the rate of change of x is 3 units per sec, 
and that of ?/ is 2 units per sec Find the rate of change of z. 

Ans. -J- = —16 units per sec 

108. Approximate Relative Rates and Errors. — The 

method of Art. 41 for finding the errors or small differences 
in a function, due to slight variations or inaccuracies in the 
independent variable, is applicable to a function of two or 
more variables. Since when an area A = f {x, y), the rela- 
tive rate of increase of A is 

dA dA 

dx , dy fj{x,y) ,fy(x,y) Uy'(A). 





A 


' A- A ' A - 


A ' 




hence, 




. , dA . , dA . 
.^^ ^dx^^ + dy ^y 


• 


(1) 


and 




^.A dA Ax dA Ay 
A '" dx A ^ dy A 




(2) 



are approximate relations. When, for example, the area of 
a rectangle is given hy A = xy, and therefore, dA = xdy -\- 
y dx, when x and y are the measurements and dx and dy the 
errors or inaccuracies, then dA gives the approximate error 
in area due to the errors dx and dy. 



APPROXIMATE RELATIVE RATES AND ERRORS 163 

If a rectangle is laid out 1000 ft. on one side and 100 ft. 
on the other, and the tape is 0.01 ft. too long; then by (1) 

AA = y' ^x-\-X' Ay = 100X0.1 + 1000 X 0.01 
= 10.00 + 10.00 = 20 sq. ft. 

is the approximate error and the exact error is 20.001 sq. ft., 
found by more laborious computation. The approximate 
relative error is by (2) 

AA^ 20 ^1 

A 100,000 5000' 

making the percentage error 
100 AA 1 



A 50 



or 0.02 of 1 per cent. 



EXERCISE XVm. 

1. In the illustrative Example 1 of Art. 107, suppose the error in 
measuring the edges was 0.02 ft., 0.03 ft., and 0.04 ft., respectively, find 
the approximate error in the volume computed with 6, 8 and 10 ft. as 
the edges. Ans. 5.32 cu. ft. 

(Exact error, AV = 5.339624 cu. ft.) 

2. The total surface of a cylinder with diameter equal to altitude is 
to be gilded at a cost of 10 cents per square inch. If the altitude is 
measured as 24 in., find the maximum error in cost, measurement being 

in. 



v^. 



3. The period of a pendulum is T = 2 tt y — . Find the greatest 

error in the period if there is an error of zLjo ft. in measuring a 10 ft. 
L, and g, taken as 32 ft./sec^, may be in error ^^ ft. per sec^. Find 
the percentage error, 

Ans. 0.0204 sec, f| per cent. 

4. In estimating the number of bricks in a pile, if the pile is measured 
to be 8 X 50 X 5 ft., and the count is 12 bricks to the cubic foot, find the 
cost of the error when the tape is stretched 2 per cent beyond the 
standard length, bricks being sold at $10 per thousand. 

5. If the side c of a triangle ABC is determined by measuring the 
sides a and b and the included angle C, show that the error Ac, due to 
inaccurate measurements, is given approximately by the equation, 

Ac = Aa cos B -\- AbcosA + aAC sin B. 



164 DIFFERENTIAL CALCULUS 

6. If the horse power of a steamship is given by the formula H = 
Kv^ D^, show that the increase in horse power, due to an increase Ay 
in the speed and an increase AD in the displacement, is given approxi- 
mately by the equation, 

AH = 3 Kv^ D^ • Ay + I Kv^ D"^ • AD. 

7. Show that the relative error in the area of the ellipse due to 

inaccurate measurements of the semi-axes a and h is given approxi- 

^ , , AA b' Aa + a- Ab 
matelyby ^ = -^ 

8. The equation for the length L and the period T of a pendulum 
being 4 tt^L = T^g, if L is calculated taking T = 1 and g' = 32 ft./sec^, 
while the true values are T = 1.02 and g = 32.01 ft./sec^, show that 
the approximate error in L is AL = 0.0326 . . ft., and the percentage 
error about 4 per cent. 

9. In determining specific gravity by the formula s = A/A — W^ 
where A is the weight in air and W the weight, find (o) approximately 
the maximum error in s if A can be read within 0.01 lb. and W to 0.02 lb., 
the actual readings being A = 9 lb., W = 5 lb., find (6) the maximum 
relative error. 

Ans. (a) As = 0.0144; 

,,, As 23 23 

(^^ T = 3600 = 36 P"''"^*- 

109. Partial Differentials and Derivatives of Higher 
Orders. — If only one of the independent variables is 
supposed to vary at the same time, by successive differen- 
tiations there are formed the successive partial differentials 
QxU, dy^u, dx^u, dy^u, ... or 

d^''^' dy^'^y' -M^"^' df'^y^' 

For example, if u = x^ + xy^ + y", (1) 

dxU = {2x + y"^) dx, dx^u = 2 dx'^, d^u = 0; 

dyU =l2xy + 2y) dy, dy'u = (2 x + 2) dy^, dyH = 0. 

If u is differentiated with respect to x, then the result with 
respect to y, there is gotten the second partial differential, 



INTERCHANGE OF ORDER OF DIFFERENTIATION 165 

For example, if u = x^ -\- x^y^, (2) 

dxU = (3 a^2 + 2 xy^) dx, dxy^u = 4:xydx dy. 

Similarly, the third partial differential dyx^^u or ^ dy dx^ 

denotes the result gotten by differentiating u once with 
respect to ?/, then this result twice successively with respect 
to X. 

The symbols for the partial derivatives are : 

d^u d'^u d'^u d^u d^u 

dx^ dx dy dy^ dx^ dy dx^ j • • • • 

In getting the successive partial differentials and deriva- 
tives of w or / {x, y) , dx and dy are treated as constants, 
since x and y are independent variables, varying by uniform 
increments. The equivalent symbols for the higher partial 
derivatives by another notation are for / (x, y) , 

JJ'{x,y), Uy'\x,y), Jy"{x,y), Sr{x,y), fyj"{x,y), 

110. Interchange of Order of Differentiation. — 

d^U d^u d^U ^ /-. 

etc. ; (2) 



dx"^ dy dx dy dx dy dx^ ' 

that is, if u is differentiated successively m times with respect 
to X and n times with respect to y, the result is independent of 
the order of these differentiations. 

It can be shown that the order is always a matter of 
indifference if fJix, y), fxy'(x, y) or fy{x, y), fyj'ix, y) are 
continuous functions of the two variables {x, y) taken 
together. 

In most cases that call for the application of the methods 
of the Calculus to physical problems the partial derivatives 
give the same result in whatever order the differention is 
done. 



166 DIFFERENTIAL CALCULUS 

For example, to verify the theorem in some cases : 

Example 1. — Given u = e^ cos y, 

du ^ d /du\ d^u 

= e^ cos y, -^^ ( ^ 1 = -^^^^-j^ = -e^ sm y\ 



dx dy\dxj dydx 

/du\ 
- ^"sm^/, -i^-^j = ^j:^^ = - e-sm y. 



— = - eMn — (—] = -^ 

dy ^' dx \dy) ~ dx dy 



Example 2. — Given u = — > 

y 

du^logz d^u ^ JI_. 

dx y ' dzdx yz 

du _ _x^ d^u _ 1 . 

dz yz^ dxdz yz' 

du _ _x log z d'^u _ log z . 

dy y"^ ' * dxdy y^ ' 

du _ logg d^u _ log 2; . 

dx y ^ dydx y"^ ' 

du _ X d^u _ x 

dz yz' dydz y'^z' 

du _ X log z d^u _ _ ^ 

. dy 2/^ ' dzdy yH 



EXERCISE XIX. 

Verify the identities (1) and (2) of Art. 110 in each of the following 
nine examples: 

1. 2i = cos {x -\- y). 2. u = e^ sin y. 3. u = cos xy^. 

4. It = x^y^ + ay^. 5. u = log (x^ + 2/^). 6. w = y^. 

7. u = xy cos (x -\- y). 8. u = tan"^ - . 9. w = sin^ x cos y. 

10. if« = (x+j,)^ ^_ + j,^-^ = _. 

■11. liu = (.' + y')K ,.g + 2.,£| + ,^0 = O. 

1 ^i^4_^_n 

12. If ^ = (^2 + ^2 + ^2)1 ' da;2 "^ dy^ "^ (^^^ "• 



EXACT DIFFERENTIALS 167 

rihj 

13. If w = e^^^ T-^rV = (1 + 3 x?/2 + xV^') u. 

ax ay az 



14. If w = sin~i (a:i/2), 



a^ 1+2 x^yH"^ 



dxdydz {\ — x^y'^z^)^ 

111. Exact Differentials. — An expression of the form, 
Mdx+Ndy, (1) 

where M and N are functions of x and ^, may or may not be 
the differential of some function of x and y; if it is, it is called 
an exact differential. Some simple expressions may be seen 
at once to be exact differentials; thus, ydx-]-xdyis an exact 
differential, for it is recognized as the total differential of 
xy. 

If M dx -\- N dy is an inexact differential, no function 
F (x, y) can be found the differentiation of which will give 
this differential; thus y dx — x dy is an inexact differential. 

In applying the Calculus to problems in physics and 
mechanics expressions like (1) frequently arise and some 
test is needed to determine whether the expression can be 
gotten by the differentiation of any function of the variables 
involved. 

As an example, the work W of moving a particle in the 
XY plane gives rise to the expression 

dW = Xdx + Ydy, (2) 

where X and Y are respectively the x- and ^/-components of 
the force acting on the particle. Since work is the product 
of force by distance, 

X = -^— and / = —j—} 
dx dy 

and (2) takes the form 

Here (3) was not gotten by differentiation of any function 
W = f{x, y) and the question is whether it could be so 



168 DIFFERENTIAL CALCULUS 

gotten. In general, if M and N are any chosen functions of 
X and y, does a function of the independent variables (x, y) 
exist that will upon differentiation give M dx -\- N dy? 
If there is such a function u = F {x,y), then 

du = -:r- d^ -\- -J- dy. (4) 

dx dy ^ ^ ^" 

Now if the differentiation of the given function gives 

Mdx-^Ndy, (1) 

a comparison with the exact differential given in (4) gives 

,, du ■J.J du . /_. 

that is, M and N must be the partial derivatives of the 
function u with respect to x and y, respectively. 

According to the theorem of Art. 110, differentiating (5) 
gives 

dy dxdy dx 

Hence, if M = -r- and N = -^ , itis manifest that 

— = -— (6) 

dy dx 

is the necessary condition that M dx -\- N dy may be gotten 
by the differentiation of a function F {x, y), and it may be 
shown that it is a sufficient condition. 

•■■ When the condition (6) is satisfied, M dx -\- N dy is an 
exact differential; when the condition is not satisfied, M dx + 
N dy is an inexact differential. 

Example 1. — Given M dx + N dy = ydx+xdy. 

TT ixr AT dM ^ dN . 

Here M = 2/, N = x, ^=1, ^ = 1. 

The condition (6) is satisfied and ydx -\- xdy is an exact 
differential. The test is hardly needed in this simple case, 



EXACT DIFFERENTIALS 169 

as it may be seen at once that the function sought is xy-\-C, 
where C is a constant, positive or negative, or zero. 
Example 2. — Given M dx -\- N dy = ydx — xdy. 

Here, since — r— = 1 and —r— = — 1, the condition (6) is 
' dy dx 

not satisfied; hence, y dx — x dy is an inexact differential, 

and no function of {x, y) exists, the differentiation of which 

will give this differential. 

Note. — If the equation ydx — xdy = is given, it may 
be changed to an exact differential equation; M dx -\- N dy = 
0, being called an exact differential equation when M dx -\- 
N dy is an exact differential. 

Thus, multiplying by y~^, the equation given becomes 
ydx -xdy _ 

which is exact, and the function F (x, y) is given by x/y = C, 
Again, multiplied by l/xy, the equation given becomes 

dx _dy^Q 

X y ' 

which is exact, and the function F {x, y) is given by log x/y = 
log C. Either of these results evidently implies the other. 
Multiplying the equation given by —x~^ gives F (x, y) by 

y/x = Ci. 

Example 3. — Given Mdx -\- N dy = -dx -\-\ogx dy. 

X 
TT Hr y l^T 1 dM 1 dN 1 

Here M = -, N = \ogx, -[- = -, -p- = -• 

X dy X dx X 

The condition (6) is satisfied and the differential is exact. 
It is easy to recognize that F {x, y) is in this case y log x. 
Example 4. — Given M dx -\- N dy = sinydx + x cos y dy. 

TJ TIT • AT ^M dN 

Here M = sm 2/, N = x cos y, -7— = cos y, -j— = cos y. 

The condition (6) is satisfied and the differential is exact. 
The function F (x, y) may be seen to be xsiny. 



170 DIFFERENTIAL CALCULUS 

Example 5. — Given x dy — y dx. Change to value in 
polar coordinates, hy x = p cos d,y = pmiO) x dy — y dx = 
p2 dd. Dividing by x^, 

xdy — ydx p^ dd 



p2 cos^ 6 



= sec 



where the differentials are exactj and the function F (x, y) is 
y/x = tan ^. (See Note Example 2.) 



EXERCISE XX. 

Determine which of the following differentials are exact, and for 
such as are exact find the functions that differentiated would give them: 

1. ymii2xdx + sin^ x dy. Ans. y sin^ x. 



2. 


(2/e^ + e^) dx + (e^ + xe^ 


)dy. 


Ans. ye^ + xe'". 


3. 


{y'-2xy)dx-\-iSxy^ - 


■x^)dy. 


Ans. y^x — x^y. 


4. 


v^dp + npv^'^dv. 




Ans. pv^. 


5. 


e^ sin ydx + e"^ cos y dy. 




Ans. e^ sin y. 


6. 


y^ 

— dx -\- X log X dy. 




Ans. Inexact. 


7. 


{x^ — y) dx — X dy. 




Ans. I x^ — xy. 


8. 


qX (^2 _j_ ^2 + 2 x) dx + 2 


e^ydy. 


Ans. e^ (x2 + 2/2). 



112. Exact Differential Equations. — Equations of the 
form 

Mdx + Ndy = 0, 

are called exact differential equations when M dx -{- N dy is 
an exact differential, the total differential of some function 
of {x, y), M and A^ being functions of x and y. The solving 
of differential equations involves the Integral Calculus, and 
the preceding Article with the Examples and Exercise are 
introductory to the subject. 

The finding of the function from which an exact differen- 
tial may be gotten by differentiation is essentially Integra- 
tion, the inverse process to Differentiation. 

In Part II on the Integral Calculus the subject of differen- 
tial equations is given further treatment. 



PART II. 
INTEGRAL CALCULUS. 



CHAPTER I. 
INTEGRATION. STANDARD FORMS. 

113. Inverse of Differentiation. — It has been shown 
that, when a function is given and its rate of change is 
required, the derivative, which expresses the rate of change, 
is gotten by the differentiation of the function. 

It often occurs that the rate of change of a function is 
known and the value of the function is desired. In many- 
problems in pure and applied mathematics the derivative or 
the differential of some function is given and the function 
itself is required. 

The derivative or the differential of a function being given, 
it is a natural inference that an inverse operation to differen- 
tiation should yield the function. This inverse operation, 
the opposite of differentiation, is called integration and to 
integrate any given function (which when continuous is 
always the derivative of some other function) means to find 
that other function whose derivative is the given function. 
The function to be found is called an integral of the given 
function, which is called the integrand; that is, a function 
is an integral of its differential. The process of finding an 
integral of a given function is integration, the inverse oj 
differentiation; that is, integration is anti-differentiation and 
an integral is an anti-differential. 

171 



172 INTEGRAL CALCULUS 

When dy = d(f(x)); d'^dy) = d--^ {df (x)) , (read "the 
anti-differential of dy equals the anti-differential of d{f{x))'' is 
the inverse expression, reducing to y = f (x), as the two 
symbols neutraUze each other. The sign of integration is, 

however, / , an elongated S; and this symbol indicates 

that the differential expression before which it appears is 
to be integrated, the whole expression denoting the integral 
itself. 

Thus Cdy^d-Hdy) and f d (f (x)) ^ d'^df (x)) ; 

the sign of integration and the symbol of differentiation 
indicating inverse operations here neutralize each other, so 



/ 



There is here a close analogy with the algebraic signs of 
evolution and involution; for example, Vx^ = x, the two 
symbols indicating inverse operations neutralizing each 
other. The analogy extends further to the fact that, while 
the operation of raising a given number to the second or 
other power is a direct operation and involves no difficulty 
in any case, the inverse operation of extracting a root may 
not be done so directly and in many cases can be done 
approximately only. While it has been shown that every 
continuous function has an integral,* this integral may not 
be expressible in terms of the elementary functions. In 
such cases, however, an approximate expression for the 
integral may be obtained by infinite series or by the measure- 
ment of an area representing the integral. Most of the 
functions that occur in practice can be integrated in terms 
of elementary functions, either directly by the knowledge 
acquired from differentiation, by reversing the rules of 
differentiation, or by reference to a table of integrals. 

* By Picard, in Traite d' Analyse. 



INDEFINITE INTEGRAL 173 

Except for simple differential expressions the process of 
integration is less simple and easy than the process of differ- 
entiation. Just as any finite number can be raised to a 
power, so can any finite continuous function be differentiated; 
and as the roots of some numbers can be expressed approxi- 
mately only, so the integrals of some functions can be 
expressed approximately only. 

There is one function whose integral is not some other 
function but is the function itself. This is the function e^, 



whose derivative is e^. As I e'^dx 



j e'^dx = e^, so Vl = 1; the 



particular analogy in this exceptional case is manifest. 
114. Indefinite Integral. — When 

dy 
dx 



f (x) or dy = f {x) dx, V = J f (^) dx, 



read ''y is equal to an integral of f (x) dx." An integral of 
dy is evidently y, and / (x) is an integral of its differential 
f{x)dx. 

Thus integrals of many simple differential expressions are 
known directly, by merely recalhng the function which 
differentiated results in the given expression. 

However, since the differential of any constant term of 
a function is zero, the function sought may contain a con- 
stant or constants no indication of which appears in the given 
differential or derivative. 

Hence, the integral of a differential expression is in 
general indefinite, owing to the lack of knowledge as to 
the existence or value, if existent, of constant terms of the 
function sought. 

If F (x) is a function whose derivative is / (x), then 



/ 



/(x) dx = F (x) -{- C, is the indefinite integral, where C 

is a general constant, called the constant of integration, de- 
noting a value either positive or negative or zero. 



174 



INTEGRAL CALCULUS 



dy 
dx 



116. Illustrative Examples. — Example 1. — When 
/' (x) = m, is given as the constant slope of y = fix); 



then y = I mdx = mx + C. The result is indefinite be- 
cause j^ = m, is the slope oi y = mx + C, y = mx — C,or 
ax 

y = mx. The constant C added to right member of the 
equation includes all constant terms, if any, of the function; 
for if the result be written y -\- C = mx + C", then y = 
mx + C" — C = mx -^C. The letter C, often omitted, 
should be written as part of the result of the integration. 

Data may be available in some cases to make the value of 
C known, or to eliminate it, and thus to make the result 
determinate. 

In this example, if it is known that the function has the 
value h when x is zero, then y = mx -^ b, since C is equal to 
h when x is zero. If y is —b, or if y = 0, when x = 0; then 
y = mx — b, or y = mx. 




As shown in the figure the function is a straight line 
making an angle ( = tan"^ m) with the Z-axis, the con- 
stant of integration being the 7-intercept. The indefinite 
or general integral is y = mx -\- C, any straight Hne with 
slope m. 



ILLUSTRATIVE EXAMPLES 



175 



Note. — When the value of C is determined the integral 
is called a particular integral. 

Example 2. — When ^ = 2 x is given as the rate of 



change of y with respect to x, then y 



/2. 



dx = x^ + C, 




where x^ + C is the general integral of 2 a; dx, since the 
differential of (x^ + C) is 2 x da:. Here x^ + C is a function 
whose rate of change is 2 x, and 
if the rate of change is that of 
the ordinate to the abscissa, or 
the slope of a curve, then the 
integral, y = x^ + C, is the equa- 
tion of the curve. The locus of 
the equation is a parabola with 
its vertex at a distance C above 
or below the origin, or at the 
origin, according as the value 
of C is positive, negative, or 
zero. 

If y is known for some value of x, then the value of C is 
easily determined. For instance, if it is known that the 
point (a, h) hes on the curve, then y = x"^ -{- C, must be satis- 
fied by the coordinates (a, h), giving h = a"^ -]- C, and, there- 
fore, C = h — a^. Hence, the particular parabola is ?/ = x^ + 
h — 0^. If the curve is known to pass through the origin, 
then, since Q is zero, y — x^ \s the parabola with vertex at 
the origin. 

dA 
Example 3. — If a given derivative -^ = 2 x represents 

the rate of change of an area A to a length x, then A = 

j 2xdx = x^ -\- C, is an area where x^ may represent the 

area of a variable triangle formed by the straight Hne y = 2x, 
the ordinate at any value of x, and X-axis. In the figui-e 



176 



INTEGRAL CALCULUS 



shown the area A is zero when x is zero, and therefore C is 
zero. The area of any triangle being one-half the product 
of base and altitude, the result of the integration, A = x^, 
is seen to be true. The area A = x"^ -\- C may represent a 
square of side-x and some additional area represented by C, 
undetermined, as it might be positive, negative, or zero — • 
the derivative of the area in either case being the given 
rate 2 x. 





dA 

Example 4. — If the given derivative is -^ =«= x^, then 



-/ 



x^dx = i^-]r Cj 



smce 



d(^^-C\ = x'dx. 



Here the area -^ is that bounded by the curve, a parabola 

y = x^y the ordinate at any value of x, and the X-axis. In 
the figure shown the area A is zero when x is zero, and there- 
fore C is zero. 

It is seen that the area 0PM is exactly one-third of the 
area of the circumscribed rectangle. Hence, the area OPN 
between the curve, the abscissa at the end of any ordinate, 
and the F-axis, is two-thirds the area of the same rectangle. 



ILLUSTRATIVE EXAMPLES 177 

Example 5. — The acceleration of a falling body being 

nearly constant near the earth's surface, it is required to 

find the velocity and the distance after any time. If s 

denotes the distance along a straight line positive upward 

dh 
and t the time, then -^ is the acceleration. 



dv d^s 



dt~dF-~dr= -3 «^ ^\dt) = ~^^^' (^) 

Integrating gives velocity, 

v = -^^ = -gl + (.C^v,), (2) 

where Vq is the initial velocity, when ^ = 0; then, 
ds= —gtdt-\-Vodt, 
Integrating gives distance, 

s= -igt'-\-vot + (C = so), (3) 

where Sq is the initial distance, when ^ = 0. If the body falls 
from rest, Vq and So are zero, hence; v = —gt; s = —igt^; 
and v'^ = —2 gs, by eliminating t. (See Art. 14.) 

Example 6. — Determine v and s in terms of t for a bullet 
shot vertically upward with a velocity of 2000 feet per 
second, neglecting air resistance. 

dv d^s 

-tT = ^ttj = —32.2 ft. per sec. per sec. 



ds 
' = dt 



I^ = r-32.2rf^ = -32.2^ +(C = Vo = 2000), 
V = Vq, when t = 0. 
s= jds= j-S2.2tdt-\-2000dt= -16.1f+2000t 

+ (C = So = 0), s = So = 0, when t = 0. 
To find the time of rising, make v = = —32.2 t + 2000; 
.-. ^ = 62.1 sec. 



178 INTEGRAL CALCULUS 

To find the height it will rise, s = -16.1(62.1)2 + 2000 
(62.1) = 62,112 ft. 
To find the time of flight, s = = -16.1 f^ + 2000 t', 

.'. t = 124.2 sec. and t = 0. 

Hence, the time of falling is the same as that of rising, 
since the time of flight is twice that of rising. The height it 
will rise may be found, by making y = in 

.^ = .-2,.; .-. . = g = M = 62,112ft., 

the same as above. 

Remarks. — These examples -illustrate the important fact 
that the knowledge of the rate of change of a quantity to- 
gether with the knowledge of its original' value, makes 
possible the complete determination of the value of that 
quantity at any time. This must be so, since two different 
quantities with the same rate of change always have a con- 
stant difference, the rate of change of their difference being 
zero. This is in accordance with the undoubted fact that 
if the rate of change of a quantity decreases to zero and 
remains zero, the quantity ceases to change at all, being 
then constant. The fact is formulated in principle (iv) of 
Art. 116, and is the converse of the fact that if a quantity is 
constant, its rate is zero. 

116. Elementary Principles. — While there is a general 
method of differentiation, for the inverse process of integra- 
tion no general method has been devised. For the integra- 
tion of the various differential expressions, rules have been 
formulated and special methods have been found, one or 
more of which provide for every case in which integration 
is possible. 

These rules or formulas are derived or disclosed through 
knowledge of the rules of differentiation; in fact, the rules 
most used are merely directions for retracing the steps taken 
in differentiation. 



— = log a; + C, since d (log x) = — ; 



ELEMENTARY PRINCIPLES 179 

Elementary principles that apply in integration may be 
expressed as follows: 

(i) ff(x)dx^F{x)-{-C, if dF(x)^f(x)dx. 

This principle furnishes the most direct proof of formulas 
for indefinite integration, and provides a decisive test of the 
correctness of the result of any integration. Thus, 

x''dx = — rT + C, since di — -^) = x'^dx; 
n + 1 V^+1/ 

/ 

In this manner the test can be apphed to prove any formula, 
or to verify the result of the integration of any expression. 

(ii) A constant factor can he transposed from one side of the 
sign of integration to the other, and a constant factor can he 
introduced on one side, if its reciprocal is introduced on the 
other, without changing the value of the integral. 

For, if a is a constant, 

jaydx = a f ydx, 

since the differentiation of the equation gives, ay dx = ay dx. 
Hence, 

I ydx = - I aydx = a I -ydx. 

(iii) The integral of a polynomial is equal to the sum of the 
integrals of its several terms. For 

/ (a'-jr x — x^)dx= I adx -\- I xdx ^ l x^dx, 

since the differentiation of the equation gives 

adx + xdx — x'^dx = adx -j- xdx — x^ dx. 



180 INTEGRAL CALCULUS 

(iv) I = C, since dC = 0. 
The integral of zero is a constant. 



Thus, if -j7 = '^i where v is constant velocity, 

-=T^ = -^ = 0; that is, acceleration is zero, 



'^^'^''^' Iit-^ I^ = ^ = "■ 



117. Standard Forms and Formulas. — There follows a 
Hst of standard integrahle forms, that is, differential functions 
whose integrals can be expressed in finite forms involving 
no other than algebraic, trigonometric, inverse trigonometric, 
exponential, or logarithmic functions. To integrate a func- 
tion that is not expressed in terms of an immediately inte- 
grate form, it is reduced if possible to one or more of such 
forms and the formula applied. 

The formulas in general are gotten by merely reversing 
the formulas for differentiation, and each can be proved by 
the principle (i) of Art. 116. 

The list will be found to contain the one or more than one 
integral to which every integrable form is reducible. These 
forms may, therefore, be called fundamental although only 
the first three are really fundamental, since each of the others 
by substitutions can be reduced to one of the three. While 
the Hst is of standard integrable forms, it may be supple- 
mented by other integrable forms; but no list of forms is 
exhaustive, even when extended into tables of integrals. 
After the acquirement of familiarity with the rules of differ- 
entiation, and the common methods of reduction with the 
standard integrals, the use of tables of integrals for the com- 
plicated forms is recommended; much time otherwise given 
to formal work in integration being thereby saved. 



STANDARD FORMS AND FORMULAS 181 

X" dx = — r-r + C} where n is not — 1. 
n + 1 

/dx 
— = log a; -h C = log x + log c' = log (c'x). 

til. fb-dx = r^ + C. 
J log 6 ' 

IV. fe''dx = e^ + C 

V. I sin a: da; = — cos x -\- C, or vers x -\- C\ 

VI. / cos xdx = sinx -\- C, or — covers x + C 

VII. / sec^ xdx ^ tan a; + C 

VIII. / CSC" xdx = — cot x + C 

IX. / sec X tan xdx = secx -jr C. 

X. I CSC a: cot xdx = — esc a; + C. 

XL / tan xdx = log sec x + C = —log cos x-{-C. 

XII. / cot xdx = log sin a; + C = —log esc x-\- C. 

XIII. I CSC xdx = log tan ^ + C = log (esc a: — cot x) + C. 

XIV. / sec xdx = log tan (0 + 7) + ^ 



= log (sec X + tan xj + C. 
'^^ = itan-i- + C, or -^cot-i- 



XVI. r^^=^log^^+C = itanli-i-+C'. {x^<a^) 
J o?—x^ 2a a—x a a 

= 1 log^±£+C = icotli-i-+C'. (a;2>a2) 



182 INTEGRAL CALCULUS 

XVII. r^^ = ^log^+C=--coth-i^+C'. {x'Xj?) 
J x^—a^ 2a x+a a a ^ 

2a a-\-x a a ^ ^ 

XVIIL J^^=^ = ^irr^l^-C, or -cos-i^ + C. 

XIX. r-4==log(a;+Vi^+^)+C,orsinh-i-+C'. 
J V x^ -\- a^ CL 

XX f / ^^ =log(x+Vi^:^)+C,orcosh-i-+C'. 
J Vx^ — a^ o, 

XXI / — / = - sec-i - + C, or — csc-^- + C. 

XXII. / /^ "^ , = vers-i ^ + C, or - covers"! ^ + C 
J V2ax — x^ ^ ^ 

The two forms of the integral in several of the formulas 
correspond to different values of the constants denoted by 
C and C. Thus in formula V, 

vers X + cos x — 1 = C — C y 
and similarly in XVIII, 

sin-i- + cos-i- = ^ = C'-C. 
a a 2 

When the differentials of two functions are equal, their 
rates are equal; therefore, the functions will be equal or 
differ by a constant; hence the variable parts of the indefi- 
nite integrals of the same or equal differentials are equal or 
differ hy a constant. 

118. Use of Standard Formulas. — When a given func- 
tion to be integrated is not expressed in a form immediately 
integrable, by various algebraic and trigonometric trans- 
formations or substitutions, the effort is made to reduce it to 
one or more of the standard forms so as to apply the formula. 
When different methods are used the results may have 



USE OF STANDARD FORMULAS 183 

different forms, but upon reduction they will always be found 
to differ (if at all) only by a constant, in accordance with the 
statement above. 

Formula I is the standard formula for the Power Form. 
It is of most frequent application, and it may be expressed 
in words as follows : 

The integral of the product of a variable base with any con- 
stant exponent {except —1) and the differential of the base is 
the base, with its exponent increased by 1, divided by the in- 
creased exponent, and a constant. 

The proof has been given in (i). Art. 116; it may be derived 
thus: since 

j2xdx = x^ + C, J 3x''dx^x^ + C, etc., 

(n + l)x''dx = x"+i + C; 
hence, in general, 



/' 



/ 



x'^dx = — — r + C'. 
n -\- 1 



When a given integrand is a fraction with denominator 
to a power, it may become this form by bringing up the 
variable quantity with change of exponent's sign; but, since 
the variable quantity is represented by x in the formula, it 
is essential that the differential of the variable quantity, and 
not merely dx, be present in the integrand before the formula 
is apphcable. 

If a constant factor is lacking, it may be supphed in ac- 
cordance with (ii). Art. 116; but it should be noted that the 
principle is only for constant factors. 

The value of an integral is changed when a variable factor 

is transferred from one side of the sign / to the other; thus, 

Cx^dx = ix^-{-C, but X fxdx = ix^ + C. 



184 INTEGRAL CALCULUS 

When a change of sign is needed, the constant factor — 1 
effects the change. Thus, for an example, 

r xdx ^1 r(a2_^2)-|(_2a;dx)= -V^^^^ + C. 
J Va^ — x^ ^ J 

To verify: 

fd{-Va^-x'-^C)= C-^ia'-xTH-^xdx) 



f 



xdx 

: , as given. 



Va2 - x^ 

When a variable factor is lacking, resort may be had to 
expansion and then application of the formula to each term 
of the polynomial. Thus, for an example : 

C2{l+x^ydx = 2 C{l + 2x^-{-x')dx = 2ix+^x^+^x')-{-C, 

When the numerator is of higher power than denominator, 
reduce by division and then apply formula or formulas, thus : 

= ^x^ -\- X + 2\og{x - 1) + C, 
If n = — 1, formula I gives a result that is not finite; but 
when n = — 1, the form reduces to form II and that formula 
applies. Thus, 

I x~^dx = j — = log a: + C. 

Formula II may be stated in words as follows: 
The integral of a fraction whose numerator is the differential 
of its denominator is the Napierian logarithm of the denomina- 
tor^ and a constant. The result will be real only when x is 
positive. When 

^>a, J ^-zr^ = log (x-a)-\- C; 
but, if 



USE OF STANDARD FORMULAS 185 

Formula III may be stated in words as follows: 

The integral of the product of a constant base with a variable 
exponent and the differential of the exponent is the base, with 
exponent unchanged, divided by the Napierian logarithm of the 
base, and a constant. 

Here the base b must be positive and not unity. 

Formula IV is the special case of III, the Napierian 
logarithm of base e being unity. 

In applying these two formulas to given integrands, it 
is essential that the differential of the variable quantity, 
and not merely dx, be present. Thus, for examples: 

fe^/" dx = n fe^/^ — = ne^/«+ C. 

The following are examples of integration by one or more 
of the first four standard formulas. 

EXERCISE XXI. 

In these examples the results may be verified by (i), Art. 116, and 
the verification should be made where the result is not given. 

2. fiax + brd. = lfia. + bradx = ^-f^±^ + C. 



x^dx 



3. C{2a + Shxydx. 4. f— 

5. /(l + ^y dx. 6. J«2 (2 _ <3)3 dt. 

7. f2Ty(^^^ + iydy. 8. fV2J^ds. 

9. CV2pxdx = V2pfx^ dx = f V2px^ ( = ! x V2px) + C. 

10. f (ax"+&)^x"-idx= — r(ax"-h&)Pnax"-idx= ^"^""^^^^'t' +C. 



186 INTEGRAL CALCULUS 

1. j5a;VTr272(ix=-f J(l-2a;2)^(-4rdx)=-|(l-2a;2)HC. 

2. JVrr7^e^dx= - J{l-e^)H-e''dx) = -f (l-e^)* + C. 

J x^ J 1 — n ' 

j~ IV j^x{\-x^)dx, 

_ r -{2ax-x^)dx _ -{^ax^-x^)^ 
' J (3ax2-x3)^ 2 

«• /x-^ -^^ = I /Ira! <*^ = '»^ (^^+ 2 -) + ^• 

9. J j5 d:, = 5(^-^ + ^46alogx-2J + C. 

20. /(logx)»f = 5^|^ + C. 21. /(log.)'f. 

22. J^dx = x-| + |-Iog(x + l)+C. 

23- /^=l°8aog.)+C. 

24. f ^ ^ + I rfx = X + log (3 a; - 1)^ + C. 

%/ o X — J. 



sin 2 X , «- /• cot X 



drc. 



26. r,-^24^dx. . 27. r 

J 1 + sin2 X J _ 

28. r(e^-e-*)2da:=-^^^-^^^^-2x+C. 29. {\e^ -\- e~A dx, 

-f—r dx. 32. (a^'b^ dx = , "^, , + C 

e^ + 1 J log a + log 

33. I -. — ^r- dx. 34. I 5 dx. 

J sin 2 a; J I ^ x^ 

«_ r logxdx _ _ 1 r —2 log a; dx 
J x(l -log^a;) ~ ~ 2 J 1 - log2 x ~x' 

* J (1 + x^) arc tan a; J tan~^ x 



DERIVATION OF FORMULAS XI, XII, XIII, AND XIV 187 

119. Derivation of Formulas XI, XII, XIII, and XIV. — 

By the application of Formula II the following results: 

*tan X sec x dx 



I tan xdx = I 



secx 

= log sec x + C = — log cos x -\-C. (XI) 

cos a; 



/cot xdx = I - — dx 
J smx 



smx 
log sin X + C = —log CSC x-^C, (XII) 



/esc xdx = I - — 
J smx 

-h 



dx 



2 sin x/2 cos x/2 



Or 

'cscx ( — cotx + CSC a;) dx 



j CSC xdx = I - 



CSC X — cot X 
= log (esc X — cot x) + C. 

j sec xdx = I CSC (x + t/2) dx 

= log tan {x/2 + 7r/4) + C. (XIV) 

Or 

'sec X (tan x + sec x) dx 



j sec xdx = j 



sec X + tan x 
sec X tan xdx + sec^ x dx 



sec a; + tan x 
= log (sec X + tan x) + (7. 

EXERCISE XXn. 

By Art. 116, and one or more of the standard formulas I to XIV. 

1 r/ • o I sr • ^\ J cos 3 a; , sin 5 a; , - x . ^ 

1. J ( sm 3 a;H-cos 5x—sm-^]dx= 5 1 = h 2 cos 5 +C. 

i% C ■ J sin^x , ^ 

2. I sm X cos xdx = — ^ f- C. 

of -J cos^a; , „ 

3. J cosxsmxrfx = 2 1"^' 



188 INTEGRAL CALCULUS 

*• /S + S?^«)'*= -/cos-«(-sin<»*) + /sin-<,cosed9. 

6. rJ^M= f'^-.t^^>do= C(c.ce-.me)cie. 

J smd J smd J ^ ' 

6. Jsin^0d0 = /5^^^^^^c^ = ia-isin(2^)+C. 

7. /cos^0d^=/i±^f-^^cZ^ = i^ + isin(20)+C. 

8. fsin^ddB = ^(1 - cos20) sin(9d0= -cosO + Icos^e + C. 

9. fsins cos3 d0 = fsin^ 6 (1 -sin2 0)dsmd = i sin* 0- 1 sin^ 0+C. 

0. jtan^ xdx = J (sec2 a; — 1) dx = tan a: — x + C. 

1. fcof^xdx = J(csc2 a; — 1) dx = —cot x — x + C. 

2. f tan3 c^ = ftan d (sec^ - 1) d^ = | tan2 - log seed + C 
= I tan2 Q _|. log cos + C. 

,3. rsec2 (ax2) x Ja; = — tan {ax^) + C. 

*• /^iS^<^^= ^/sec(a.^)tan(ax')d(ax^)= isec(ax»)+C. 

B. r^^?^= r^2|£^= rcsc(2x)d(2x) 
J sm X cos a; J sm (2 x) J 

= log tan .T + C, by XIII. 

6. I- = rcsca;seca;da;= (- c/a;=logtana:+C,by II. 

J sm X cos X J J tan x ° ' *^ 

7. f sec2 6 csc2 Odd = C (sec^ + csc^ ^) d0 = tan - cot + C. 

/cot + tan ,^ c r^r. 1^ 1 1 X /« . 'r\ . ^ 

9. fl^^d«= r(i.=4af)!d9 = 2(tan9-sec9)-9 + C. 
J 1 + sm J cos2 ^ 

20. fesinx^^Qg^^^ 21. Je'^^'^'sinxdx. 

22. J6*^^(''"^sec2(aa;)c;a;. 

120. Derivation of Formulas XVI, XVII, XIX, and XX. — 

By the application of Formula II, the following results: 



DERIVATION OF FORMULAS XVI, XVII, XIX, AND XX 189 
For XVI, put 

(^ — x^ 2a\a-\-x a — x)^ 

r dx ^ _i_ r dx _ j^ r -d 

J o} — x^ 2aJa-{-x 2aja — 



dx_ 

X 



= 2^ log (a + x) - 2^ log (a - x) + C 

1 , a-\-x . ^ 

= ---\og \-C 

2a a — X 

= - tanh-i ^ _j_ c\ (x^ < a') (XVI) 



or 



r dx ^ r -dx _ 1 r dx 1 r_dx_ 

J a" - x^ ~ J x"" - a" ~ 2 aj X + a 2aJ x- a 

= ^log^i-^ + C = icoth-i - + C. (x2 > a2) 
2a X — a a a ^ 

The first or second of these results is used according as 
a — X or X — a is positive; that is, the form of the result 
which is real is to be taken. 
For XVII, put 

1 - W 1 L_Y 

x^ — a^ 2a\x — a x -\- a) 

/ dx ^ I C dx I r dx 

x^ — a^ 2a J X — a 2aJ ~^x -\- a 

= 2^ log {x-a) - 2^ log {x-{-a)-\-C 

1 , X — a . ^ 

= TT log — i h C 

2a x-\- a 

= - i coth-i - + C\ (x' > a^) (XVII) 
a a ^ ^ / V / 



or 



r dx ^ r -dx ^ j_ r -dx _ j_ r dx 

J x^ — a^~ J a^ — x^ 2a J a — X 2aja + x 

= ^log^^+C= -itanh-i- + C'. (x2<a2) 
2a ^a-\-x a a ^ ^ 



190 INTEGRAL CALCULUS 

The first or second of these results is used according a.sx — a 
or a — a; is positive. 
For XIX, let 

Vx" -f a2 = z-x; or z = x + Vx"" + a% (1) 

/. a^ = z^ — 2 xz. 
d{a?) = = 2zdz-2xdz- 2zdx; 
{z — x)dz = zdx; 

dz dx dx 



z z-x Vx2 + a2 
/. r^L== ff = log. + C = log(x + ViH^) + C, 

or sinh-i - + C. (XIX) 



a 
For XX, similarly, on letting Vx^ — a^ = z — x, 



I 



^^ =log(x+Vx^-a^) + C, or cosh-i- + C'. (XX) 

The logarithmic form of cosh-i - is log ( ^ "r ^ x — a \ ^ 

but its derivative or differential is the same as that of log 
{x + Va:^ — of), the constant a disappearing in the differ- 
ed 
entiation; and so too with the sinh~i-. (See Art. 66.) 

a 

121. Derivation of Formulas XV, XVIII, XXI, and XXII. 

— These formulas are merely the reverse of the differential 
forms given in Examples 1 and 2, Exercise VI. 

They may be derived from the forms for the inverse trigo- 
nometric functions of a;. Thus: 
/f dx 
dx 1 I a _ 1 / "^ Va/ _ 1 . -\^ \ri 



smce 




DERIVATION OF FORMULAS XV, XVIII, XXI, AND XXII 191 

Since 

tan-i -=l- cot-i -, d f tan-i -] = d(- cofi -V 
a 2 a^ \ aj \ a) 



Hence 



/; 



^^ =-tan-i-+C, or -icot-i- + C'. (XV) 



0? -\- x^ a a a a 

In the same way the second forms follow for formulas XVIII, 
XXI, and XXII. 

The standard forms are given in terms of -, because they 

are of more use than those in terms of x; the latter, being 
special cases where a = 1, are often given as the standard 
forms. Integrals may be obtained by reduction to either 
form. 

EXERCISE XXin. 

1 C dx _ 1 f d {ex) _ 1 ex ^ 

J 62 + c2a;2 c J 62 + {exY he 6 ^ ^* 

he 



= c-/r^^^ = 2i'-S^ + ^ (^^^^>^^) 



be 



r dx r dx _ 1 ,„^-i 3; + 3 ^ 

Ja:2 + 6x + 5 J (x + 3)2 - 4 4 ^ (a; + 3) + 2 ^ "" 

- r x^dx 1 , x^ — 1 , ^ ^ r dx 

„ r xdx 1 , _, ^2 o r dx 



192 INTEGRAL CALCULUS 

10 r ^^ = 2 r ^^^^ 

J aa;2 4- 6a; + c J (2 ax + 6)2 + 4 ac - 62 



tan 



^ - — + C (4 ac > IP-) 



y 4 ac - 62 V 4 ac - 62 

1 , 2ax + 6- V62-4ac , „ ,. ^ , „, 
log + C. (4 ac < 62) 



V 62 - 4 ac 2 aa; + 6 + V 62 - 4 

11 f ^^ + ^ dx-- f (2^ + 4)^^ , o f ^^ 

Jx2 + 4x + 5 J a;2 + 4x + 5"^ J (x + 2)2 + l 

= log (a;2 + 4a; + 5) + 3tan-i (x + 2) + 0. 

J a;2 + 2x + 1 ~ J (x + 1)' J (a; + 1)2 

= log(x + l)+^-i-^ + e.. 

^n C dx \ r h dx 1 . , 6a: , ^ 

J Va2c2 - 62^2 J V(ac)2 - (6a;) 2 ^ ' «c 

1 , 6a; , „, 

or — 7- cos 1 \- C. 

ac 

14. C—=M= = i log (6a; + Vb^^+~^^) + C, or sinh-i — + C. 

16. f , "^^ = I log (6a; + V6^^^"3"^) + C, or cosh-i — + C\ 
-^ V62a;2 - a2c2 o ^ ac 

16.- r^J^= = 4. r , ^" = 4-- sin- X \/^ + c. 

-^ Va - 6a;2 Vb -^ Va/b - x^ Vb ^ « 

17. f-J^= = -l^sin-^x\/l + C. 18. r-7^^- 
-'V3-2a;2 V2 '^ 3 -^ V3 - 4a;2 

19. f ^ ^^ r ^ ^^ ^.^.-i2^jti + r- 

-^ Vl - X - a;2 J Vf - (x + i)2 V 5 



20. r^=== = sin-1 ^V-^ + C- 
•^ V 2 + 2 a; - x2 V 3 

21. /^=^== 1 r 



Vax2 + 6x + c Va "^ V(2 ax + 6)2 + 4 ac - 62 



4^ log (2 aa; + & + 2 Va Vax2 + 6a; + c) + C. 



22. r , = log (x + a + Vx2 + 2ax) + C. 

^ V x2 + 2 ax 



, EXERCISE XXIII 193 

23. C—M= = -i^log (xV^+Vax'-b) + C. 
J y/ax^ — b V a 

nA f dx 1 r 2adx 

24. 



, r dx _ _!_ r 

^ V- ax^ + 5x + c Va*' 



V4 ac + 62 - (2 ax - b^ 



1 . , 2ax — b , ^ 
^- sm-i - + C. 

V a V 4 ac + 62 



J VI - X *^ V 1 - a:2 J Vl - a;2 •^ 

= sin-i a; - Vl - x^ + C. 



•^ V a; — 1 -^ V x2 — 1 

dx c cdx _ 1 _, ex 

ex V(cx)2 - (a6)2 a6 
5 da; 



-_ /- da; f c dx 1 , ex , ^ 

27. I , = I 7=== = -rsec 1^ + C. 

»^ X vc2x2 - a^¥ -^ ex v (cx)^ — {abY ^o ab 

28. / 



X V 3 x2 - 5 

29. f ^±l rfx= r^±^dx = sec-i-+log(x+V^^i3^)+C. 
^ X Vx—a ^ X vx2 — a2 « 

30. f^^ZEZrf^= r '']Z^ dx= C^dx^_r_^dx^ 
J X J X V x2 - o? -^ V x2 - a2 •'' X V x2 - a2 

= Vx2 - a2 - a sec-i - + C. 
a 

31. r_J^= r_^:i^= r(x-2-i)-ix-3(ix=-^=:+c. 

J (1 _ 3;2)i J (a;-2 _ 1)^ J Vl-x2 

ort f dx If bdx 1 , 6x , ^ 

J V2 abx - 62x2 6 J V2 a (6x) - (6x)2 6 a 

„„ r — dx 1 , 6x , ^ 

33. I — = r covers^ -r + C. 
J V 8 6x - 62x2 6 4 

34. f ^ ^^ =vers-^— + C. 
•^ V ax - x2 « 

xdx fa — 2x — a , r(a — 2x)dx a r dx 



„_ r —xdx _ r a — 2x — a , _ r {a — 2x)dx a r 
^ Vax - x2 »^ 2 Vax - x2 ' -^ 2 Vax - x2 2 J 



Vax — x2 



2x 
= V ax — x"-^ — TT vers~^ 

36 



Vax — x2 — - vers ^ \-C. 

2 a 



/dx r dx . ./2x — a\ . ^ 

, = I , = sin-i I — - — j + C 

Vax - x2 ^ Vay4: - (x - a/2)2 \ a / 



194 INTEGRAL CALCULUS 

Note. — It may be noticed that the result for Example 34 differs 
from the last result for Example 36. The difference is accounted for 

by the values of the constants of inte- 
gration. As may be seen in the figure, 

sm 1 1 1 I + - = vers ^ — , 

\ a / 2 a 



1 


r 


;;^ 


1 x^ 






// 




\-¥-}^ 


^^ 


( 


1 




1 .-^ 


\^ 


1 




/ 


t/^ 


\x 



that i 



IS, 

arc BP + ^ = arc OBP. 

o\^ 2x :ma' ^ 

a=2.:radius=l Each result may be verified by dif- 

ferentiation according to (i) Art. 116. 
These results illustrate the statement at the end of Art. 117. 

122. Reduction Formulas. — A formula by which a 
differential expression not directly integrable can be re- 
duced to a standard form or a form easier to integrate than 
the original function, is called a reduction formula. 

A general reduction formula that has a wide application 
and is most useful in the reduction of an integral to a known 
form is the formula for integration by parts. 

Many special formulas of reduction are obtained by apply- 
ing this general formula to particular forms. 

123. Integration by Parts. — Differentiation gives 

d (uv) = udv -\-v du. 
Integrating, 

uv = Id (uv) = I udv -{- I vdu; 

transposing, 

I udv = uv — I V du. (1) 

The formula (1) may be used for integrating u dv when 
the integral of v du can be found. This method of integra- 
tion by parts may be adopted when / {x) dx is not directl}^ 
integrable but can be resolved into two factors; one, as dv, 
directly integrable ; and the other, as v du, a standard form 
or a form less difficult than u dv. 



INTEGRATION BY PARTS 195 

No rule can be given for choosing the factors u and dv 
other than the general direction that the factor of / {x) dx 
taken as dv is first chosen as that part directly integrable, 
and then what remains whether one or more factors must be 
taken as u. 

When the function given to be integrated contains more 
than one factor that is directly integrable, there is some 
choice to be exercised in selecting the factor dv, and in some 
cases a different choice may be necessary, if the first choice 
results in v du being non-integrable. It may be that one or 

more applications of the formula to I v du will be effective. 

The use of the formula is illustrated in the following 
examples, the formula being written, 

I f{x)dx = I udv 

= uv — j V du. (1) 

sin~i -dx = X sin~i - + v a^ — a;^ + C 

X 

Let dv = dx : then u = sin"^ - > 

a 

, dx 

V = X, du = 



V a2 - x^ 
Substituting in (1) : 

/> t X 1 • f X I X OjX 

sin~i -dx = X sin~i I , 
a a J Va2 - x^ 



= a;sin-i - + V a2 - x"" + C. 

0/ 



(Compare example in Art. 118.) 
Example 2. — j X' cosxdx = X'sinx — j sinxdx 



= a; sin a; + cos a; + C, 



196 INTEGRAL CALCULUS 

Example 3. — 
I x^ sin xdz = 2x sin x — (x^ — 2) cosx + C 

I x'^-smxdx= cc^ (— cos x) +2 cos X' xdx 

= a;^ ( — cos x) + 2 (x sin x+cos x) + C, by Ex. 2, 
= 2 a; sin a; — (a;2 — 2) cos x + C. 
Example 4. — 

a;Mog xdo; = —^^ (logo: - ^^^J + C, 

I logiC'ii;''ax= — —r 'logx— / — r— • — 
J n + 1 ^ Jn + la; 

/y. n+1 /* /« n+1 

-logx- J ^_ , ,,, + C 



n + 1 ^ J (n + l)2 

.n+1 



= 7-^ (log X — r ) + C. 

n + lV ^ + 1/ 

Example 5. — / a;e^ c/o: = xe"" — e^ + C. 

I e'^'xdx = e'^-ix^ — i j x^-e'^dx. 

The last form is not so simple as the original, indicating 
that a different choice of factors should be made. Another 
choice gives 

j X'B^dx = X'e"" — j e^'-dx. 

= xC" — e^ + C 
Example 6. — 

fVa^-x^dx = ^xVa^-x'' + ^sin-i- + C, (1) 

/Va^ — x^ • da; = Va^ — a;^ • a; + I , 
*J V a2 - a;2 

= X Va^ — a;2 -{- / , dx 

^ V a^ — x^ 

= a; Va2 - a;2 + a^ sin-i - +C' - fVa^'^^^dx. 
a J 

Transposing the last term and dividing by 2 gives (1). 



INTEGRATION BY PARTS 197 

Example 7.* — 
fV¥To^dx==^xV^+a^+^anog{z+V^+a^)-\-C (2) 

= i X V¥+a^+ I a" sinh-i - +C\ (2') 

Example 8.* — 
CV^^^dx= ^xVx2-a2-ia21og(x+Vx2^^)+C (3) 

= lx V^^^- i a^ cosh-i - -\-C'. (30 

Note. — The integrals of the three last examples are of 
sufficient importance to be considered as standard forms. 

Example 9. — 
I X ( — jdx = I X' sinh xdx = x cosh x — sinh x -{- C. 

I X • sinh xdx = X' cosh x — I cosh a; • do; 
= a; cosh x — sinh a; + (7. 

EXERCISE XXIV. 

Verify the following by j udv = uv — i vdu. 

1. fcos-i -dx = x cos-i - - Va2 - x^ + C. 
J a a 

2. ftan-i - (Za^ = a: tan-i ^ _ 1 log (^2 + a;2) + C. 

3. f cot-i - (^o; = a; cofi - + i a log (a2 + x^) + C. 

J Cb Oj Z 

4. J log a; c?a; = x (log a; — 1) + C 

5. \x log a; da; = I ^2 (log a; — J) + C 

6. J a;3 log a; (ia; = \x?' (log x — |) + C 

* By same method as in Example 6. 



198 INTEGRAL CALCULUS 

7. Cx (e«^ + e-«^) dx = - (e«=^ - e""^) - \ (e«^ + e"'^*) + C. 

8. f e^ sin x da; = | e^ (sin x — cos a;) + C. 

Take w = e^ and apply the formula, then take u = sin x, apply 
formula; add results. 

9. 1^2 cosxdx = 2x cos a; + (a:^ — 2) sin x + C. 
10. j (log x)2 (^a: = a; [(log x)^ - 2 log a; + 2] + C. 

11. fx^iios.yd. = ^ [(logx)^ -^ttt'o^^ + (^J +^- 

12. r^x^ tan-i X = (a; - i tan-i x\ tan-i x- log (Vl + x^) + C. 

124. Reduction Formulas for Binomials. — By applying 

the formula for parts to j x"^ {a + hx^'Y dx that integral may 

be made to depend upon a similar integral, with either m or 
p numerically diminished. 

There are four such formulas, which are useful for refer- 
ence, but there is no need that they should be memorized. 



/ 



x"^ (a + bx'^y dx = —7-7 — V-^ — , : . 
[np + m + 1 ) 



_ a{m — n -{- 1) 
h {np + m + 1) 
or 



/ x'^-^ (a + hx'^y dx, (A) 



np + m + 1 Tzp + m + U 



or 



a(m+l) a(m+l) J 

or 

an(p+l) ari(p+l) J 

Formulas (A) and (B) are used when the exponent to be 



REDUCTION FORMULAS FOR BINOMIALS 199 

reduced, m or p, is positive; (A) changing m into m — n, 
and (B) changing p into p — 1. 

Formulas (C) and (D) are used when the exponent to be 
reduced, m or p, is negative; (C) changing m into m + n, 
and (D) changing p into p + 1. 

When any denominator becomes zero the formula is in- 
applicable, and the integral can be obtained by some method 
without the use of reduction formulas. 

Formulas (A) and (B) fail when np + m + 1 = 0. 
Formula (C) fails when m + 1 = 0. 
Formula (D) fails when p + 1 = 0. 

EXERCISE XXV. 
1. f f""^^ = sin-i - + C, when m = 0, Standard Form XVIIL 
rxd^^ r(^2_3.2)-^a;^a;= _Va2-x2+C, whenm = l. (1) 
x'^dx r^,. „x-4 , a:"^i Va2 - x^ 



r , ^ = ( x"^ {a^ - x^y^^ dx 

im-XWr^^^^^ by (A). 






+ ^ sin-i - + C, when m = 2. (2) 



rx^^ = fxs (a2 - x2)-^ (ia: = - I Va2 - a;2 
»^ Va2_a;2 J 6 



+ o a^ I , , when m = 3, 

3 J Va2 - x^ 

= - I' Va2 - a;2 - |a2 Va^ - x^ + C, by (1). (3) 
o o 

CJ^M= = Cx' (a2 - a;2)-^ dx= -"^ V^^IT^ 
J Va2-x2 ^ ' ' 4 

, 3 , f x2 (^a; , . 

+ - a2 I ^ , when m = 4, 

4 J Va2 - x2 

4;"^T^/ "^4^^^^ '-+C'»by(2). (4) 



200 INTEGRAL CALCULUS 

2. I x"^ va^ - x2 dx = — r -TT I , , by (B). 

J m + 2 m-\-2J Va^ — x^ 

r V^TT^e^a; = ? V^i":::^ + |' f . "^^ , when m =0, 

•/ ^ I J Vn2 _ .V.2 



2 ^^ ^ + 2 '''' a + ^- V 6, Art. 123. ) 
(x^ V^rr^^a; = ^^^'~^' + ^ (^M=., when m = 2, 

= (f - 1^) V^^^^ + 4^ sin-i ^ + C, by (2) of Ex. 1. 

3. r-^Ii^ = |V^Mr72_|log(^ + V^r+^2)+c, by(A), 

- £ V^M^2 _ ^' sinh-i - + r' /"Compare Ex. \ 
-^^x-\-a ^sinh ^ + C. \^ 7^ Art. 123. / 

4. f /^^ =fV^rr^ + |'iog(a;+V^nr^) + c, by(A), 

= ^V^^Tr^ . ^cosh-i-+C' /Compare Ex. 8,\ 
2 vx a + 2 cos^ a+ ^ • V Art. 123. ) 

5. I x^ va:2 ± a2 c?a: = ± -r I , , by (B), 

J 4t 4: J a/t2 -Ur72 



(j±|^)V^±^-^log(x+ V^^±^)+C, byExs.3,4. 



r^m+i Vx2 ± a2 a2 



(B). 



6. I x"* V ic2 ± a2 da; = -^r ± — p^ I ^==, by 

J w + 2 m + 2J Vic2 ± a2 

7 f ^^ _ Vrc2 ± a2 

Ja;2Vx2=fca2~^ o?x "^ 

Jx^ (x2 ± a2)"^ dx, by (C). 



=Fa2 
(-1 -2 + 2 + 1 



=Fa2 



I r_^^___ = _ ^^^ ~ ^^ 4- C 
•^ a:2 Va2 - x'^ ^^^ 

Jx-^ (a2 - xT" dx = "'"'^l""''^' 

- rl(-l-2^+2 + l) j^ (,. _,.)-!,,, by (C). 



REDUCTION FORMULAS FOR BINOMIALS 201 

/x-Hx2-a2)~^dx = ^^^^^|^^+ 2^ Jx-Hx^-a^r^cZa;, by (C), 
= ^"^1"/!^' + A sec-i - + C, by Standard Form XXI. 



/da: 1 1 _, re , „ 
J = , r sec 1 - + C. 

I x^ (x2 — a2) * do: = ^ — - — 

- i ^x-^ (^' - o')~^ dx, by (D), 
= -L=r - ^sec-1- + C. 



11, r ^^ _ ^ _ _|_ Q^ 

*^ (a2 - x2)2 a2 Va2 _ x^ 

C( 2 2^-s A x{a^ - x^y^ 

J(a2-x2)fd^=- 2^,^_,^ 

12. rV2ax-x2 da: = ^^ V2ax-x2 + |' sin-^ ^-^^ + C, 



or 



— ^r— V 2 ax — x2 + —vers ^ - + C. 



Or 



^ ^ \ in succession. / 

r V2 ax - x2 dx = rVa2 - (x - a)2 dx 



X — a 



V2 ax - x2 + ^ sin-i ^ — - + C, by Ex. 2, 

^ ^ a 

^ - « ./Fi 5 1 «^ _i a? , ^, / See Ex. 36, \ 

= „ V2 ax — x2 + — vers ^ - + C. ( -^ . vvttt ) 
2 2 a \Exercise XXIII. / 

13. (x^ V2ax-x2 dx = f x^l V2a-x dx = - ^"""^ (^^-^ - ^')^ 

<2m + l)a r ^_, V2ax-x2dx, by (A). 
m + 2 J 



202 INTEGRAL CALCULUS 

x"^ dx x^-^ V2 ax - x^ 



"•/ 



15./ 



V2 ax-x^ w 



. (2m - l)a r x'^'^dx , ,,, 
+ ^^ — I / , by (A). 



dx V2 



aa; — x'- 



X- V2ax -x^ .(2m- 1) ax'' 

m — 1 /• da; 



■ m — 1 /• da; , ,^. 

(2m-l)aJ x"'-iV2ax-a;2' "^ 



da: Va2 



^ xW a^ - x^ 2 a2x2 2 a^ Va2 _ a;2 + a 



r _w 2 ,^_i , a;-3+i (a2 - a;2)^ 
Jx3(a2-a:^).da; = ^,^-3-p3y- 



-'^-':lt''-'^ f^-ia^-^^)-dx, by(C), 



Va2-a;2 , 1 f dx o tt 10 



dx Vx2 + o2 1 



C. 



17 C ^^ = _ ^^ + a^ 1_ , X 

•^ x3>/x2 + a2 2 a2x2 2 a^ ^ _^ \/x2+ a^ 

Jx^(x2+.a2).dx= ,.(13^1) 

Vx2 + a2 1 f dx o TT in 

= --2^i^-2^2J ^V^Hr^ ' SeeEx.l9. 

18. r ^^^ =llog-— ^ +C. 

•^ X V a2 - x2 « V a2 - x2 + a 

Here m + 1 = 0, therefore Formula (C) fails. 

Let a2 - x2 = ^2; /. -a;dx = 2d2, x2 = a2 -22. 

/ dx _ r dz 1_, a — z ^ 
X Va2 - x2 "'^ z^ -a^~ 2a ^^a + z'^ 

1 , a - Va2 - x2 , ^ 

= TT- log , + C 

2a ct + Va2-x2 

= -log ^ +C. 

» a + V a2 - x2 



REDUCTION FORMULAS FOR BINOMIALS 203 
19. f ^j^_ = llog ,-^^ +C. 

Again m + 1 =0, and Formula (C) fails. 

Let a2 + x2 = ^2; .-. X c^a; = 2 cZs, x^ = z^ - a?, 

J X Va;2 ^a^~ J z^ -a^~ 2a ^^2 + a"^ 

= jr- log . h C 

2a ^ Va;2 + a2 + a 

= -log ■ ^ + C. 

a Va;2 + a2 + a 

A^oie. — The integrals of Examples 18 and 19 may be considered as 
additional standard forms. 

„^ fVa2-x2 , ,/^— , , 1 ^ , ^ /By (B) and\ 



CHAPTER II. 



DEFINITE INTEGRALS. AREAS. 

125. Geometric Meaning of / / (x) dx. — As the repre- 
sentation of an integral by an area between a curve and an 
axis is of fundamental significance, and as the effort to find 
an expression for the area of plane figures bounded by 

curved lines gave rise 
/ to the Integral Calcu- 

lus, such representa- 
tion will be given 
further treatment 
than illustrated in the 
examples of Art. 115. 
Let P2OP1 be the 
locus oi y = f{x), and 
let the area between 
the curve and the 
"■^' ic-axis be conceived as 
generated by the va- 
riable ordinate MP or y, as the point (x, y) moves along the 
curve and x increases. Let A denote the area bounded by 
the a;-axis, y = f (x), some undetermined fixed ordinate as 
MoPo or M2P2, and the moving ordinate MP. 

Let Ax = dx be MMi; then, while A A the actual incre- 
ment of the area A is MPPiMi, dA is MP DMi, the incre- 
ment that A would get if, at the value MqPqPM, the change 
of A became uniform and so continued while x increased 
uniformly from the value OM to OMi. Hence 

204 




J^'^^M^ 



DERIVATIVE OF AN AREA 



205 



dA = MPDMi = ydx=f{x) dx, 
/. A = I ydx = I f(x) dx, 

where A is indeterminate so long as the fixed ordinate MqPq 

or M2P2 is indeterminate. 

dA 
126. Derivative of an Area. — Since dA = y dx, -j- = y; 

that is, //je derivative of the area with respect to x is the ordinate 
of the hounding curve. 

This important result may be obtained by the method of 
limits also, taking the increments infinitesimal. Thus, 

AA = MPPiMi, 

AA > 2/ Ax, and AA < (?/ + Ay) Ax, 
.-. yAx<AA<{y + Ay) Ax, 
AA 

dA ,. A A 
•*• "1— = hni -T — = V, 

dx Aa;=0 Ax 



smce 



lim (y + A?/) 

Ax=0 



2/, Ai/ = as Ax = 0. 



In case y decreases as x increases, 
the curve falls from P to Pi, and the 
inequahty signs are reversed, but the 
result is the same. 

Let A be the area between the i/-axis 
and the curve; then, 



dA ,. A A 

-r- = lim -T — = x, 

dy Ay-o Ay 



or dA = xdy, 



I xdy. 




dA'NPBNl 



Here, the derivative of the area with respect to y is the abscissa 
of the hounding curve. 



206. INTEGRAL CALCULUS 

127. The Area under a Curve. — Let the curve y = f (x) 
of Art. 125 hey = x\ 

Let OMq = a and OMi = h; then 

MoPoPM = A= Cx'dx = ^ + C. 

As the area is measured from x = a, 

... A = = | + C, C=-|, 

.-. A. = S-f^ (1) 

where A^is the variable area MqPq PM. Making a; «= 6 in 
(1) gives 



a 



MoPoPiMi = ^6 = I - |- (2) 

The usual notation is 

128. Definite Integral. — In general, when h > a the 
increment produced in the indefinite integral F (x) -{- C by 
the increase of x from a to 6 is 

F(h)+C- {F(a) + C) ^F{h) -F{a). 

This increment of the indefinite integral of f{x) dx is called 
*Hhe definite integral of /(x) dx between the limits a and 6," 

and is denoted by I / (a;) da;. Hence, 



X 



'f{x)dx'=F{h)-F{a). 

a 



The operation is that of finding the increment of the indefinite 
integral of/ (x) dx from x = ato x = h, where h is called the 
wpper or superior limit, and a the lower or inferior limit, 
although they are more precisely termed ''end values" of 



DEFINITE INTEGRAL 207 

the variable, as they are not ''hmits" in the usual sense of 
the word. If the upper end value is variable, then 

P/ (x) dx = F (x)T =F(x)- F{a) . 

When the lower end value a is arbitrary, —F (a) may be 
represented by an arbitrary constant C, hence 



X 



f{x)dx = F{x)+C. 
Since 

^ f{xy_dx=:F{x)-\-C, 



I' 



an indefinite integral is an integral whose upper end value 
is the variable and whose lower end value is arbitrary. 

Hence, when the integral is represented by an area and 
the area is known for some value a of x, 

A = Cf{x)dx = F{x) -F{a), 

where C is —F (a), and the area A is determinate. 

If the area under the curve y — x^ (Art. 127) be reckoned 
from X = 0; when x is zero, A is zero, therefore, C is zero 

and A^ = -^, the area of 0PM, (3) 

(See Example 4, Art. 115.) 

Ay = fy^ dy = ly^ = l x\ the area of OPN. (30 

Making x = am (3) gives 

Aa = -1^, the area of OPqMq, (4) 

o 

and making x = 6, gives 

¥ 
Ab = -^, the area of OPiMi. (5) 

o 



208 



INTEGRAL CALCULUS 



As definite integrals, 

If, in Example 3 of Art. 115, the area A is from x = a to 



or 



3 Jo 



X = b; then 



A = r2xdx = x^\ = h^-a^ 



the area of a trapezoid. It may be noted that, when the 
integral is '' between Umits," it is not customary to write 
the constant, as it will be eliminated. 

129. Positive or Negative Areas. — The area under or 
above a curve y = f (x), from x = a to x = h, will be posi- 
tive or negative according as y is positive or negative from 
X = a to X = h; hence, when the curve crosses the x-axis, 
the areas are gotten separately, otherwise the result will be 
the algebraic sum of the areas and may be zero, since the 
areas above and below the axis may for some curves be equal. 
For example, the areas for the curve of sines or of cosines 
as shown in Art. 140 illustrate the principle. 

130. Finite or Infinite Areas — **Limits" Infinite. — 
From the geometrical meaning of an integral it follows that 
/ (x) dx has an integral whenever / (x) is continuous, hence 

the end values a and h are in 
general taken so that/ (x) will be 
finite, continuous, and have the 
same sign, from x = a to x = h. 
lix = b = 00 , then 




/ (x) dx = lim I / (x) dx, 

a 6=00 K/ x=a 



X where the Hmit nfiay, or may notj 
exist. When the limit of the in- 
tegral is finite the total area is found; but if as b becomes 
infinite the integral becomes infinite, then no limit exists and 
the area \x^tox = b becomes infinite as b becomes infinite. 



FINITE OR INFINITE AREAS — LIMITS INFINITE 209 

Example 1. — 

A= I -^ c^:c = lim / -r c^a; = lim =1. 

Jl X^ 6=00 Ji x^ 5=^ L xji 

Limit exists although as x approaches zero, f(x) = — be- 

x^ 
comes infinite. 

A = I ~^dx = hm I ~dx = ]im\ \ , 

Jo x^ b=oo Jo x^ 6=00 L ^Jo 

which does not exist, since = oo ; that is, the area 

up to a; = 6 becomes infinite as b becomes infinite. 
A = I -zdx = Hm / —dx = ]im\ L 

Jo X^ a=oJa X^ a=o\_ xja 

which does not exist, since = oo , the area becoming 

xjo 
mfinite as x approaches zero. 




-2.0 -IS -1.0 -0.5 

Example 2. — 



1.0 1.5 



A = I C'dx = lim / e^'dx = Hm e^ = e^ 

J —CO a = — QO Ja o= — oo|_ Ja 

is total area under curve up to ordinate at P {x, y). 



210 INTEGRAL CALCULUS 

Area to right of y-axis 

= OMPB = A= f e^dx = e^f = e^ - 1. 
Jo Jo 

Area to left of y-Sixis 

= 1 e'^dx = lim l e'^dx = lim e^ =1. 

tJ —<Xi a= — 00 tJ a a=—oo ]_ Ja 

Note. — When y = f (x) = e^, y, the function, is the ordi- 
nate, equals the slope at the end of the ordinate, and may 
represent the total area under the curve up to the ordinate. 
(See Art. 138.) 

131. Interchange of Limits. — Since the definite integral 



X 



'f(x)dx = F(b)-F(a); 



it follows that 

Jf(x)dx=- I f (x) dx, 

since the second member is — [F (a) — F {b)]= F(h) — F(a). 
It follows also that the definite integral is a function of its 
limits, not of its variable; thus 



X 



/ (y) dy has the same value as / / (x) dx. 



each being F(b) — F(a). 

The algebraic sign of a definite integral is changed by an 
interchange of the limits of integration, and conversely. 

132. Separation into Parts. — A definite integral may be 
separated into parts with other limits or end values. Thus, 

P/ (x) dx = pf (x) dx + fV (x) dx. (1) 

Let the curve y = / (x) be drawn and the ordinates^Pi, 
EPi, CP2, be erected at the points for which x = a, x = b, 
X— c. 



MEAN VALUE OF A FUNCTION 



211 



Since area APPiB -= area APP2C + area CP2P1B, (1) 
follows. 




c ^ c 

If OC = c', and C'Ps is the corresponding ordinate, then, 
area APPiB = area APP3C' - area BPiPsC; 
and hence 

rf(x)dx^ rf(x)dx- r s{x)dx 

= f f W dx + Pf (x) dx, by Art. 131. 

*J a *J c' 

Note. — It may be seen that 

Jf(x)dx= I f(a — x) dx, 

for each is F(a) — F(o). Thus, 

- Cfia -x)d{a-x)= -F{a- x)T = F{a)-F (0) 

= / f{x)dx, 

133. Mean Value of a Function. — The mean value of 
/ {x) between the values / (a) and / (6) is 



tJ a 



dx 



b — a 



Let area APPiB represent the definite integral I / (x) dx. 



212 
Then 



J f (x) dx = 



INTEGRAL CALCULUS 



areaAPPi5 



= area of a rectangle with base AB and height 

greater than AP but less than BPi 
= AB . CPi 
= (6 — a)f{c), where OC = c. 



Hence 



f(c) 






x) dx 



where/ (c) is the mean value off (x) for values of x that vary 
continuously from a to 6. 




The mean value may be defined to be the height of a 
rectangle which has a base equal to 6 — a and an area equi- 
valent to the value of the integral. 

Example 1. — To find the mean value of the function Vx 
from X = 1 to X = 4:. Let OPPi be the locus of 



y 



f(c) 



Vx, 


OA- 


= 1 and 


OB 


= 4. 




£■' 


Ux 


_i 
3 


>- 


1] = 


14 


4- 


1 


9 



= 1| = 1.551 = CP', mean value, 
c = X = (-V^)2 = -W- = 2.42 = OC. 



= - = 0.6366. 



EVALUATION OF DEFINITE INTEGRALS 213 

Example 2. — To find the mean value of sin as varies 
from to 7r/2, or from to tt. 

I sinBde -cos<9 

7r/2-0 ^ V2 " ^2 """ TT 

I sinddd — cos ^ 
^^^^ ^r- = J-0 = f = 0.6366. 

TT — U TT X 

Example 3. — To find the mean length of the ordinates of 
a semi-circle of radius a, the ordinates for equidistant in- 
tervals on the arc. 



£' 



sinddd — acos^ ^ 

= - J^ = — = 0.6366 a. 



TT — TT 

Example 4. — To find the mean length of the ordinates of 
a semi-circle of radius a, the ordinates for equidistant inter- 
vals along the diameter. 



r Va' - ^ ^^ . „ _ . . 

J -a 1 ^/-7 ^ , a^ . .xf Tra^ 1 

7 r = -xVa^ - a;2 + - sm-i - = -— x ^c- 

a — (— a) 2 2 gj-a 2 2a 



a;2 dx 

= - X Va^ — a: 

2a 



= 7a = 0.7854 a. 
4 

134. Evaluation of Definite Integrals. 
EXERCISE XXVI. 

1. fVrfx^^T^ ''"^'-^' - 

./a n + 1 J a 71 + 1 

2, J\x^dx = x^T =1Q-1 = 15. 

3. ^''(f - 2^) = log a; + log (2 - a;)][^ = log {2x- x^) j' 

= log (2 a; -x2). 



V2r - 2/ 

256 Trb^ 
-6 a* '^ " ' ^'^ 315 a4 ' 



214 INTEGRAL CALCULUS 

Jr-oo 1 ~1°° 1 

a Jo a 

TT TT TT 

X^ sin /"^ ■ ~l ^ /- 

— — dd = I cos-2 d sin d0 = sec ^ = V 2 - 1. 
J cos2 Jo Jo 

6. I , = arc sm - = -• 

Jo Va^ — x^ ^Jo "^ 

„ r"" dx 1 , a;"!'' TT 

7. I -7-; — ^ = - arc tan - = -j— 
Jo a^ -{■ x^ a a Jo 4 a 

8. f' -; ""^^ =8r. 
Jo 

9. f liiy'-h'^Ydy 

10. I :j— ^ — r = ^arctana;2 = -. 
Jo 1 +^* 2 Jo 4 

11. I :i— ; — ^ = ai'c tan e^ = arc tan e^ — -r 

Jo 1 +e2^ Jo 4 

^« r^ • o^ 7. f^l-COS20,^ TT 

12. j^ s^nHdB^j^ 2 ^^ = 4* 

13. (Kos^edS^ f\±^llde=\- 

Jo Jo ^ 4 

Note. — Considering the areas between the axes and the graphs: 

IT TT 

sin" a; dx = I cos" x dx, where n is positive, 

Jo 

TT 

J sin" a; da: = 2 | sin" a: dx, where n is positive, 
"^O 

TT 

J cos" a: da; = 2 | cos" x dx, if n is an even integer, 
-^o 

but = 0, if n is an odd integer. 



135. Areas of Curves. — As has been shown, the formu- 
las in rectangular coordinates are 

A = j ydx and A = I xdy. 



AREAS OF CURVES 



215 



(a) Let A denote the area between the curves y = f (x) 
and y = F (x); let a; = OM, dx = PE; then, the variable 
area A = PoP'P and dA = PP'DE = (/ (x) - F (x)) dx; 



area 



PoPTiP = A= r\f{x)-F (x)) dx, 



where the points of intersection are (xo, 2/0) and (xi, 2/1). 

If the locus oi y = F (x) is the a;-axis and Xo and Xi are a 
and h, this formula reduces to 



/ S{x)dx. 





(b) In polar coordinates, 

A = ijp'dd. 

For let Po be any fixed point (pc, ^0) and P (p, ^) any variable 
point. 

Consider the area PoOP to be generated by the radius 
vector p as increases from ^0, and denote it by A. With 
OP as a radius draw the arc PD and let dd = Z. POPi) then 

dA = OPD = ip'pdd = ip^de, 
the increment of ^1, if uniform, as in a circle. 

/. A = i fp^dd, or A = i Cp^de, if ^0 = 0. 



216 INTEGRAL CALCULUS 

By method of limits, increments infinitesimal: 

I p2 A(9 < AA < i (p + Ap)2 A^, where AA = OPPi, 

. dA ,. AA 1 , . A . r^ \n ' n 

.*. -^ = lim -rr- = ^ p% smce Ap = as A0 = 0. 

eta A0=o t^t/ Z 



EXERCISE XXVII. 

1. Find the area between the parabola i/^ = 4 x, the x-axis, and the 
ordinate at any value of x; from x = 1 to x = 4=. 

2. Find the area between the parabola a:^ = 4 ?/, the 2/-axis, and the 
abscissa at any value of y; from y = 1 to y = 9. 

3. Find the area between the two curves y^ = ix and x"^ = iy. 

4. Find the area between the cubical parabola 4 ?/ = x^, and the 
a;-axis from a; = to x = 2; from x = Otox = — 2; from x = —2 to 
x = 2. 

5. Find the area of the semi-cubical parabola 4 z/^ = x^, bounded by 
the double ordinate at x = 4. 

6. Find the area between the line y = x, and the curve 4 2/^ = x^, 
in the first quadrant. 

7. Find the area included between the parabola x* + 2/^ = a% and 

o? 
the axes of X and Y. Ans. tt* 



8. Find the area bounded by the curve 2/ (1 + x^) = x, and the line 
y = I X. Ans. log 4 — |. 

9. Find the area included between the parabola x^ = 4 ay, and the 
witch 1/ (x2 + 4 a2) = 8 a\ Ans. (2 tt - f ) a^. 

10. Find the area bounded by the witch ?/ (x^ + 4 a^) = 8a^ and 
its asymptote y = 0. 

Area = 2 I „ , . „ = Sa^ arc tan ^r— =4 xa^. 
Jo a;2 + 4a2 2aJo 

11. Find the area bounded by the hyperbola xy = \, its asymptote 
2/ = 0, and the lines x = \ and x = n. Ans. log n. 

When n = 00 , log n = 00 ; hence the limit does not exist, and the 
area between the hyperbola and an asymptote is infinite. Since the 
area is the Napierian logarithm of the superior limit, Napierian loga- 
rithms are sometimes called hyperbolic logarithms. 



AREAS OF CURVES 217 

12. Find the area of the lemniscate p^ = c? cos 2 Q, 

4.q2 y.¥ 

Area = -^ | cos 2 ^ d^ = a^. 

^ «/0 

13. Find the area of the cardioid p = 2 a (1 — cos 6), 

Area = 2a2 f ''(I - coae)^dB = Qira^ 

14. Find the area of the circle p = 2 a sin ^. 

Area = 2a^ r''sin2 Odd = ira^. 
16. Find the area of the circle p = 2 a cos 0. 

Area = 2a^ f'cos^ddd = -n-a^. 

«/o 

16. Show that the difference between the areas of the two circles 
above, from 6 = to d = 7r/4, is a^; also that the area of one circle 
intercepted by the other is twice the area of the first circle from «= 
to = 7r/4. 

17. Find the area of the part of the circle 

p = a sin + 6 cos 6, from 6 = to 6 = 7r/2. 

18. Find the area of one loop of the curve p = a sin 2 0. 

IT IT 

Area = ^ I a^sm^2ddd = ^ i {l-cos4d)dd. Ans. ^' 

Z Jo 4 */o o 

19. Find the area between the first and second spire of the spiral of 
Archimedes p = ad. 

Area = ^ d'^dO -^ i d^ dd -= 8aV. 
2 J2t 2 Jo 

20. Find the area of one arch of the cycloid x = a {6 — sin 6), 
y = a{l — COS0). 

X2 7ro /»2ir 

ydx = a^\ (1 — cos oy dd, 
Jo 

(when X = 0, 6 = 0; and when x = 2Tra, 9 = 2t, and dx = a (1 — 
cos d) dd) 

= a2 f '^ (1 - 2 cos + cos2 d) dd = Sira^; 
Jo 

that is, the area is three times that of the generating circle. 



218 INTEGRAL CALCULUS 

21. Determine the area of the circle x^ -^ y^ = a^. 
Area = 4j ydx = 4 f Va^ — x^ dx. 

Using the parametric equations of the circle, x = a cos 6, y = a sin. d, 
where 6 is the variable parameter, gives dx = —asinddd. Sub- 
stituting the values of y and dx gives: 

Area = 4 f Vd^ - x^ dx = -4 f a?- sin2 d dd, 

I 

I when a; = a, = 0; x = 0, ^ =%) 

■IT 

a2 sin2 d0, by Art. 131 



= 4 2 f- - EEI^I^ /Compare Ex. 14 above\ 
12 4 Jo V and Ex. 12, Art. 134. / 

= '7ra2. 

To get J Va^ — x^ dx, the indefinite integral; let a; = a sin ^; dx=a 
cos0d0; then, 

J Va2 — x2 da; = a2 J cos^ ^ d(j), where ^ is the complement of d above, 

= f J(l+cos20)(^0 

= f (0+ i sin2<^) + C = I (0 + sin(^cos0)+ C 

= fsin-^ + |V^^^^ + C, 

the indefinite integral. Compare Ex. 6, Art. 123. 

Area = 4 C Va^-x^ dx=4[^ sin-i -+i V^^+ cl ° = Tra^, as above. , 
•^0 L^ a 2 Jo 

Corollary. — Area of Ellipse = 4 - f Va^ — x^ dx = 4 - ^ = 7ra&. 



TO FIND AN INTEGRAL FROM AN AREA 



219 



136. To find an Integral from an Area. — An integral 
may be found from an area, when it can be gotten geomet- 
rically from the figure. 

Example 1. — Find / Va^ — x^ dx from the figure of the 

circle y = Va^ — x'^. 

Area = BOMP = BOP + OMP = ia^ + ixy 

= -pr sin~i - + -X V a^ — x\ 
2 a 2 




M A 

If the initial ordinate is not OB and is undetermined, then, 
Area = / Va^ — x^ dx 



|Va2-x2 + |-sin-i- + C, 



as above, C being independent of x and indefinite when the 
initial ordinate is undetermined. 

Example 2. — Find / V2ax — x'^dx, using circle 

y = V2 ax — x^. 

Area = OBPM = OBPC + PCM 

1 ^^ . X- a 
= 2^^ + ~2~^ 



a^ ,x,x — a 



V2 



ax 



220 
or 



INTEGRAL CALCULUS 



Area = CBPM = BPC 4- PCM 



sm 



-1 



X — a , X — a 



V2ax-x\ 




If the initial ordinate from which area is reckoned is 
undetermined, then 



/ 



V2 ax — x^ dx 



X — a 



1 



^ V2ax-x' + ^a2 sin-i - — - + C, 
2 2 a ' 



7ra^ 



where C = ^ , if Area = 0, when x = 0; 



or 



X — a 



^ V2ax - x" + ^a2 vers-i- + C 



where C = 0, if Area = 0, when a: = 0. 
As may be seen in the figure, 



sm- 



x — a , TT 



+ ;i = OCP = vers-i - ; 

a 2 a 



that is, 

a^ / . .x — a , Tr\ a^ . 
2K^ + 2J=2^' 



x — a . ira^ a^ 



4--r = Trvers"^-' 
a 4 2 a 



(See Note at end of Exercise XXIII.) 
Either result gives 

^2 ax — x'^dx = \Tra^ = areaof 05A. 






AREA UNDER EQUILATERAL HYPERBOLA 221 



Example 3. — Find / (mx + h) dx, by means of line 
= mx + h. 

Area = OMPB = BDP + OMDB 
= ix' mx -\- X'b 



mx^ 



hx. 



If the initial ordinate is riot OB, and is undetermined, then 



/ 



mx' 



mx -\-b) dx = -^ + 6a; + C. 




137. Area under Equilateral Hyperbola. — As in the 



figure of the circle y = Va^ — x^, 



f 



Va^ — X? dx = -^x Va^ — x^ + ^ a^ sin~i 



X . 



expresses the area BOMP and a^ sin"^ - is represented by 
twice the area of the circular sector BOP] so 

V^^T^^dx =lx \/^M^2 _|_ 1 a' sinh-i - (Ex. 7, Art. 123) 
'0 ^ Z a 

may be shown to express the area AOMP under the equi- 






lateral hyperbola y = vaM-^> and a^ sinh-i - to be repre- 
sented by twice the area of the hyperboUc sector AOP, 



222 INTEGRAL CALCULUS 

To get 
I Va^ +x^dx; let x = asinh^, dx = acosh0d0; 

then, 

/ "^ VoHP^ dx = a2 fcosh^ d0 = |- (</) + sinh (^ cosh 0) 

= io; Va^ + x2 + 1 a^sinh-i -, 
2 2 a 

as also in Ex. 7, Art. 123. 




If X = a cosh 4) and dx = a sinh d0 be substituted in 
/ Vx'^ — a^dx; 



then, 



X 



Vx^ — o? dx = ^x ^x^ — a^ — — cosh~i - 



1 X 

(as in Ex. 8, Art. 123), and ^a^ cosh-i- will be represented 
by the area of a sector of the equilateral hyperbola 

y = Vx'^ — a^. 



SIGNIFICANCE OF AREA AS AN INTEGRAL 223 

138. Significance of Area as an Integral. — The units of 
the number represented by A as the measure of an area will 
depend upon the units chosen for the abscissa and the 
ordinate. If the unit for x be one inch and that for y be ten 
inches, then a unit of A would represent ten square inches; 
if on the graph the unit of x is one-tenth of an inch and the 
unit of y is one inch, these representing one and ten inches 
respectively, an area on the graph of one-tenth of a square 
inch will represent the area of ten square inches. 

The integrals represented by areas may be functions of 
various kinds, such as lengths, surfaces, volumes, velocities, 
accelerations, weights, forces, work, etc. Hence, the physi- 
cal interpretation of the area will depend upon the nature 
of the quantities represented by abscissa and ordinate. 

(a) If, in the figure of Art. 125, the ordinate represents 
velocity and the abscissa represents time, then the area 
represents distance; and, since velocity 

ds 

' = 11 = "^' 

where a is acceleration, 

dA 

—rr=v = at. 
dt 

Hence, A= f atdt = hat^ + C = MJ'^PM, 

and since s = \ af^ -\- Sq, C is So, initial distance or area; and 
the number of square units of A ( = MqPqPM) will equal the 
number of linear units of distance passed over by a moving 
point in the time t = MqM. 

(h) If the ordinate represents acceleration and the ab- 
scissa still represents time, then the area represents velocity; 
and since acceleration 

_dv dA _ _ dv 

""'It' W'^^'dt' 



224 INTEGRAL CALCULUS 

Hence, 

A= fdv= Cadt = at-\-C = MoPoPM, 

where a is constant acceleration, and since v = at -\- vo, C is 
Vq, initial velocity or area; and the number of square units 
of A will equal the number of units of velocity acquired by a 
moving point in the time ^ = MqM. 

(c) If the ordinate represents a force acting in a constant 
direction, and if the abscissa represents the distance through 
which the force has acted, then the area A = MqPqPM 
will represent the work done by the force acting through the 
distance represented by MqM. If the force is constant in 
magnitude as well as in direction, the area will be a rectangle, 

dA r 

since -z- = F, constant, gives A = j Fds = Fs -{-C. 

Whether the force is constant or variable the area 

A = j F ds represents the work done, the area being that 

under the graph of the equation y = f (F), representing the 
force. If the force is not constant in direction, the area will 
still represent the work, provided the ordinate represents 
the component of the force along the tangent to the path of 
its point of application. 

By means of certain contrivances the curve y = f (F) may 
be plotted mechanically by the force itself, as, for example, 
in the steam engine by means of the indicator. Having the 
curve, the mean force may be easily found; it is given 

ds MqM, the area divided by the distance through 

which the force acts. 

The area may be read off at once by the polar planimeter, 
and the work done found directly. 

It is manifest that a function may be represented graphi- 
cally either by the ordinate of a curve or by the area under a 
curve; if the ordinate is made to represent the function, the 



by/F 



AREAS UNDER DERIVED CURVES 



225 



slope of the curve is the derivative of the function; if the 
area under a curve is taken to represent the function, then 
the derivative of the function is the ordinate of the curve, 
since the ordinate is the derivative of the area. 

It is usually preferable to represent by the ordinate * that 
which in the investigation is mainly under examination; 
therefore, if this is the derivative, the latter method, where 
the area is the function and the ordinate is the derivative, 
should be used rather than the former method, which is to 
be used when the function is mainly under consideration. 
The function e^ is exceptional, in that the ordinate repre- 
sents the function, the slope of the curve, and the area under 
the curve. (See Example 2, Art. 130.) 

139. Areas under Derived Curves. — It has been shown 
(Art. 84, figures) by drawing the graphs of a function and 
its successive derivatives that the variation of the function 
is exhibited to advantage. 




2 0' 2 



It may now be seen that the area under any derived curve 
is represented by the ordinate of its primitive curve. Thus 
the area under the graph of y = f (x) is represented by the 
ordinate oi y = f{x), that under the graph oi y = j" {x) by 

* Irving Fisher, A Brief Introduction to the Infinitesimal Calculus. 



226 INTEGRAL CALCULUS 

the ordinate oiy = f {x), and so on for the successive derived 
curves. 

/v»3 

. Drawing the graphs of y = f(x) = -^and y = /'" (x) = 2 

together with those oi y = f (x) = x^ and y = f {x) = 2 x 
(shown in Examples 3 and 4, Art. 115), it is seen that the 
areas are represented as stated. 

It may be seen also that I f{x)dx= j x'^dx = -:^ = A, 

being represented by the ordinate of 2/ = -^, is an integral 

function of x^ and the graph an integral curve of x'^. 

If 2/ = 2 be the fundamental curve, then y = 2 x is the 

first integral curve; y = x'^, the second; y = -^, the third; 

x^ 
y =^ j^j the fourth; and so on. 



CHAPTER III. 

INTEGRAL CURVES. LENGTH OF CURVES. CURVE 
OF A FLEXIBLE CORD. 

140. Integral Curves. — li F (x) has / (x) for its deriva- 
tive, then F(x) is called an Integral Function or simply an 

Integral oi f{x). The General Integral is / / (x) dx = F (x) 

+ C, called also the Indefinite Integral. 

The graph of an integral function is called an integral 
curve. If the original or fundamental function is 

y=f(x), (1) 

then y = F(x) (2) 

is the first integral curve of the curve (1), where F (x) is that 
integral of/ (x) which is zero when x is zero. In the general 
figure of Art. 125, F (x) is the area 0PM under the curve 
y = f (x)', in the figure of Art. 139, ii y = f (x) = x^, then 
F{^x) is the area under y — x^ and is the ordinate of the 

integral curve 2/ = o"* 

It is manifest that for the same abscissa x^ the number 
that indicates the length of the ordinate of the first integral 
curve is the same as the number that represents the area 
between the original curve, the axis (or axes for some func- 
tions), and the ordinate for this same abscissa. Hence, the 
ordinates of the first integral curve may represent the areas 
of the original curve bounded as stated. 

It may be seen also that for the same abscissa x^ the number 
that expresses the slope of the first integral curve is the same 
as the number that measures the length of the ordinate of 

227 



228 



INTEGRAL CALCULUS 



the original curve. Hence the ordinates of the original 
curve may represent the slopes of the first integral curve. 
. The integral curve of the curve of equation (2) is called 
the second integral curve of the original curve of equation 
(1). The integral curve of the second is called the third 
integral curve of the original curve (1), and so on. Thus 
for any given curve there is a series of successive integral 
curves.* 

The function cos 6 and the first and second integral curves 
are shown with their graphs. 




Let y = cosd be the fundamental function, then 
y = I cosedd = sine -\- C, 
where C is zero, as y is zero when d is zero; and 
y = I sine de = —cosd -\- C, 



* The statements in the three paragraphs above with some difference 
of words are given in Murray's Integral Calculus, where a fuller treat- 
ment will be found in the Appendix. 



APPLICATION TO BEAMS 229 

where C is one, as y is zero when 6 is zero. Hence, 
y = sind and y = 1 — cos 6 = vers d 

are the first and second integral curves of the curve y = cos 6. 

It is seen that the ordinate of the first integral curve at 
e = 7r/2 is +1, that number being the same number that 
measures the area under the fundamental curve for the same 
abscissa; the ordinate being zero at ^ = tt indicates that the 
algebraic sum of the areas of the fundamental curve is zero 
and hence that the area below the axis from 6 = 7r/2 to t is 
exactly equal to that above from ^ = to 7r/2; the ordinate 
being zero again a,t 6 = 2 w indicates that the areas of the 
fundamental curve above and below the axis are exactly 
equal up to ^ = 2 tt. 

It is manifest that the ordinates of the second integral 
curve indicate the corresponding areas for the first integral 
curve, the number being +2 from ^ = to tt and 2 — 2 = 

up to ^ = 2 TT. 

In the case of this function the series of integral curves 
can be extended indefinitely without any difficulty. 

It is manifest also that the ordinate at any point on the 
fundamental curve gives the slope at the corresponding 
point on the first integral curve, the ordinate for the first 
gives the slope of the second, and so on. 

The subject of successive integral curves has useful appli- 
cation to problems in mechanics and engineering. 

Illustrative examples follow, showing the appUcation to 
the expression and graphical representation of the shearing 
force and bending moment throughout the length of a loaded 
beam; also to the slope and deflection of the elastic curve, 
the curve of the mean fiber of the material of the beam. 

141. Application to Beams. — As given in the Mechanics 
of Beams, the Vertical Shear at any section of a loaded beam 
is the algebraic sum of the vertical external forces on either 
side of the section, and the Bending Moment is the algebraic 



230 INTEGRAL CALCULUS 

sum of the moments of those forces about a point in the 
section. The moment of a force about a point is the product 
of the force and the length of the perpendicular from the 
point to the hne of action of the force. (See Art. 172.) 

The Elastic Curve is the curve assumed by the mean fiber 
along the axis of a longitudinal section through the centers 
of gravity of the cross sections of the beam, the Slope is the 
slope of the tangent to the curve at any point, and the 
Deflection is the ordinate at that point. 

Taking abscissas to denote as usual horizontal lengths, 
the ordinates will represent the quantities to be depicted by 
the curves. The fundamental curve is the curve for L, the 
load; the Shear S is represented by the first integral curve; 
the Moment M by the second integral curve; EI upon the 
Slope m, by the third integral curve; and EI upon the De- 
flection (?, by the fourth integral curve, where E and / are 
constants denoting the Modulus of Elasticity and Moment 
of Inertia, respectively. 

Example 1. — Let a beam of length I between supports 
be simply supported at each end and loaded with a uniform 
load of w lbs. per hnear ft. 

L = y = —w, w taken with negative sign as a downward 
force. 

S = y = I —wdx = —wx 

-\- (Sq = -^,S being zero when ^^ = o) • 

M = y= I (-^ - wxjdx = Y^ — ^ + (^0 = 0, ikf when 



x = 0). 



wl „ wx^ 



M (m - 2/) = J (|-x - ^j da; = 

+ f mo = ~ 24 ^^^' ^ when x = 0, m being zero at rr = ^J 



EI 



id = y)-f(^ 



APPLICATION TO BEAMS 



231 



6 24. 



dx 



12 



wx^ wlH 



- 24" - -24" + (^0 = 0, d at a; = 0). 



EXAMPLE 1. 



LoadLine 



IlilllllliniMiriniiiri^^ 



--^niminiixQjjjj^^ 



ShearLine 




Inflexion Poini d=-^^,mca:. Inflexion Point 



XDefleciion 
Curve 



384 EI 



'^^^^liil lLlllllilMlllllllimi lLUU-^ 



J,/i\i/ 



rnilllllMMIIIinTTTnT-r. 



<-Hy^J/2-->i^ -ysV5i=o.58-t -y^ii-yj-yfj/i' 

0.2LJ ; ^-rrmTTTTTTTrmTTT^ ' ^■^^'"^ 



J 



■XSlope 
CiiTue 




M:=y24iifi^TTiax. 
M=-yj2 wll max.Tveg. 
Shear and Load Lines the same as Example 1. 




XMoment 
Curve 



232 INTEGRAL CALCULUS 

Example 2. — Let a beam of length I between supports be 
fixed at each end and loaded with a uniform load of w lbs. 
per linear ft. 

L = y = —w. 

S = y = —wdx=—wx 

-\- [So = -K- ) S being zero when x = -)• 

7,^ C l^^l \ 7 "Wl WX^ 

wP 
+ (Mo, M at X = 0), Mo = - To > s^^ below. 

+ Moa; + (mo = 0, m at X = 0) 

wl wx^ ivdi \ 

= -rX^ ^ — {■^x= —MqXU from m = at x = L 

"" "12' ~ 24 ~ "^4~ + (^0 = 0, c^ at a: = 0). 

Example 3. — Cantilever Bridge.* In the cantilever 
bridge the joints are placed at the inflexion points, where 
the Bending Moment would be zero if the bridge were con- 
tinuous over the whole span. 

Let the beam of Example 2 have joints at the inflexion 
points, and let the length of each cantilever arm be denoted 
by a and the length of the suspended span by h. 

The Shear hne and the Moment curve will be unchanged 
but the Slope and Deflection curves will not be continuous as 
they were without joints in the beam. For slope and de- 

* The essential features of this example are given by H. E. Smith 
in his " Strength of Material." 



APPLICATION TO BEAMS 



233 



flection each part is to be considered as a separate beam, the 
arms having in addition to their uniform load a concentrated 
load at their jointed ends, equal to half the uniform load on 
the middle span h, since that part is supported at the joints 
by the arms. 



idiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiinmiiHiiiiiiiiiiiiiiiimiiiimii^ 



I 




<&-rsy2j^i 



i/^Yzi ->te7^^-> 



0.151 

L = —w. 



0.71 



0151 



S= C-wdx= -m + (5o = |(2a + 6), 
S being zero when x = ^J' 



w 



wx^ wa 



From M = = 2 (2a + &) ^ - ^ - -^(« + fe), 

X = a and x = a -\-h, 
that is, 

x=(l-iV3)Z/2 and x = (l + i Vs) 1/2, 
and 
6 = 1 V3 ?, from M = Q ^ -^ 2'~12^ Example 2. 



234 INTEGRAL CALCULUS 

It is seen that the maximum negative moment tV wV^ at 
the fixed ends is twice the maximum positive moment ^5 wP 
at the middle. For equal strength, the bending moments at 
the ends and middle of the beam should have equal numeri- 
cal values; this requires that 

wa , , , . wly^ 

that is, 

4a2-l-46a = &2. or 4a2 + 4fea + 6^ = 2fe2; 

or (2a + 6)2 = 262, 

/. 6 - J V2I and a = {l - \ V2) 1/2. 

It follows that 

_ = _ and --(a + h) = -j^, 

giving equal strength at ends and middle of beam. 

Example 4. — Let a beam of length I between supports be 

fixed at the right end and simply supported at the left end; 

and let it be loaded with a load uniformly increasing from 

zero at the left end to w lbs. per linear ft. at the right end. 

J w 

L = y = - j^' 

/w w 

— jxdx = — ^x"^ -{- {So =S, when x = 0). 

+ {Mo = 0, M when x = 0). 
EI {m = y) = f(s^ - ^^x^dx = ^- ^x^ 

+ [ Wo = "oT TT- , m at o; = J , from m = at a; = Z. 

S/(d = 2/) = j (24 - ^ + ^ - 24^^7^ 

= 24^~ ~2~^^~Q' ~ i20] + (^o = 0,dwhena; = 0). 



M = y = I (So — ^x^] dx = SoX — ^'x 



APPLICATION TO BEAMS 



235 



EI{d^y) = 
wl 



wl^ Sd' , S,P wl' 



24 



+ 



6 



120 



-,, when X = I, 



/. So = jx, supporting force at 0. 
. S = v = ---x'l =--wl 



Si at right end; supporting force = + jrwl. 



°r"''""'''''™^™mimmMMS^M§S 



' iniiniiiiiiiiii — 






— ^mrnimiinjj^ 



Shear Curue 



irnmnTTTTrr. 



M=^0.05wp,max. 



at right end, M= 0. 06 % wl^ max. neg. 




y3VS^5t=0.V7-t >| 



LoadLine 



Moment 
Curve 



X 

Slope Curve 



^""""m^ 



■jppj^^— 



XDeflection 
Curve 



d= 



0.916 wl'^ 
384 FT 



'^ ^ 10 2i^ ' 
/. X = I a/5 I = 0.447 . . . Z, when the shear is zero. 

V5 



,, wl wx 

M = y= z^x 



DX^~\ _ 

6i Jx=iV5Z 



rs 



wP = 0.03 wP, 



236 INTEGRAL CALCULUS 

maximum positive moment where shear is zero, since 



^ = s = o. 

ax 



TIT ^^ 



-^L=-3^^'^=-«-°«^"''^' 



maximum negative moment. 

T,, „ wl wx^ 
^ = ^=10^-^' 
/. x = V§l = i V3X 5 = 0.77 , . . I, 
where moment is zero and where there is'an inflexion point 
on the Elastic Curve, the fourth integral curve, since M is 
second derivative of d with respect to x. 

EIim = y) = -^ + ^^- 24;J^^^= -j^,EI upon slope 
at left end. 

120"^ 20 Wr 
.*. X = i Vs Z and I, when slope is zero. 
TIT / 1 \ wl^ , wlx^ wx^ 1 0.016 /- ,. 

^/ (<i = 2/) = - 120- + -60- - l20]L^r - -1^ "^^ "' 

= -0.002385 wP = -^^rf,J^/upon max. deflection. 

OOTt 

The deflection is zero at x = and x = Z, as used above. 

142. Lengths of Curves. — Rectangular coordinates. — It 
has been shown in Art. 10, (d) that ds^ = dx^ + dy^; now let s 
denote the length of the arc whose ends are the points 
(xo, 2/o) and {x, y), then ds = Vdx^ + dy^; whence 

or 

-£[-(1)']'* 

according as ds is expressed in terms of x or of y. 



LENGTHS OF CURVES 237 

In getting the length of any curve, that formula which 
gives the simpler expression to integrate is the preferable one 
to use. 

EXERCISE XXVm. 

1. Find s of the circle x"^ -\- y- = a^, and the circumference. Here 

dy _ _x^ (dyV _ ^ . 
dx~ y' \dx) ~ y^ ' 

hence ^ ^ TS} ^ "'] ^^ ^^ ^ f ^^' "^ "^'^^ ^^' ^i^S (D' 

J'^ dx . .x~\^ . ,x . ,xq 

xo Va2 — x^ ^-1^0 ^ ^ 

Circumference, s = 4 a sin~i - = 4 - a = 2 xa. 
aJo 2 

2. Find s of the semi-cubical parabola ay^ = a;^. Here 

\dx) ~ 4:a* 

From the origin, s = ^y [(^ + 4I) " ^J • 

3. Find s of the cycloid x = a arc, vers - =F V2 ay — y^. Here 

(dxV ^ y . 

\dy) 2a -y' 

:. s = VYa f^ (2 a- y)~' dy, using (2), 
''2/0 

= 2 V2^ [(2a - 2/0)^ - (2 a - yf]. 

Making 2/0 = and y = 2 a and taking twice the result gives 8 a for 
the length of one arch. (See Ex. 3, Art. 97.) 

Note. — Finding the length of a curve is called rectifying the curves 
since it is getting a straight line of the same length as the curve. The 
semi-cubical parabola was rectified by William Neil and by Van Heu- 
raet also, and it is the first curve that was absolutely rectified. The 
second rectification, that of the cycloid, was by Sir Christopher Wren 
and by Fermat also, and the third was of the cissoid by Huygens. 
These rectifications were effected before the development of the 
Calculus. 



238 INTEGRAL CALCULUS 

4. Find s of the parabola a;^ = 2 'py, {xo, yo) being the origin. Here 
fdyV^o^, 
\dx) p2' 

= - [I V^^+^ + ^ log {x +Vp2+^2)JJ [Ex. 7,Art. 123.] 
= 2^V2^ + x» + |log- ^ 

6. Find s of the catenary y = a cosh - = |(e" + e "j. Here 
:. s = J I 1 +1^6" -2 + e « jj cZa; 

6. Find s of the ellipse 2/2 = (1 - e^) {o? - x^), and the length of 
the curve, e being the eccentricity. Here 

dy ^ -(1 -e^)^x . /dyy ^ (1 - e^) x"" . 
dx Va^ — x^ ' \<^a:/ a^ _ 2;2 ' 

J* 1 dr 

{o? - eH^Y —^ — ; (1) 

a;o Va2 - a;2 

hence, the length of the elliptic quadrant Sg is 

Sg = C (a2 - eH^)^ , ^^ • (2) 

For the integration of (1) and (2), see Ex. 5, Art. 203 and Note. 

7. Find s of the circle, and the circumference, using the parametric 
equations x = a cos d, y = a sin 6. Here 

fdx\^ sin2 d , 

I T" I = — v~n ; dy = a cos Odd; 

\dy/ ■ cos2 ^ ' ^ ' 

do = a{d -do) \ =2 ira. 

-J Jo 



LENGTHS OF POLAR CURVES 239 

More directly; 

nO ~l27r 

ds = add; :. s = a\ dd = a {d - do) \ =2x0. 
JOq Jo 

8. Find s of the involute of the circle, and length of the arc between 
6=0 and 6 = ir, using the equations 

X = a (cos 6 -\- dsind), y =a (sin 6 — cos 0). 
(See Cor. Ex. 2, Art. 97.) Ans. | aB^; | ar^ 

9. Find s of the cycloid, and the length of one arch, using the para- 
metric equations x = a {6 — sind), y = a {I — cos 6). Here 

dx = a{l — cosd) dd; dy = a sind dd; 

:. s = f (dx^ -\-dy^)' =a^2 f (1 - QOse)^dd 



^^0 



A Q 

sin - dd, since Vl — cos = V2 sin - 



= -4ac0S2j^^ = 4«(^C0S2-C0S2JJ^ ' 
.'. Z = 8 a, for one arch. 

10. Find s of the parabola y"^ = 2 px, (xo, 2/0) being the origin,' and 
the length of the arc from the vertex to the end of the latus rectum. 

Z = p/2[V2 + log(l + V2)]. 

11. (a) Find s of the hypocycloid x^ -\- y^ = a^, and the length of 
, the curve. (a) Ans. fa* {x^ — xo'); 6 a. 



^ ' ^ (6) Ans. 4 I a sm^ = 6 a. 

143. Lengths of Polar Curves. — It has been shown in 
Art. 77, (3), that for a polar curve ds^ = p^ dd^ + dp\ Now 
let s denote the length of the arc whose ends are (po, ^0) and 
(p, d), then ds = Vp^dd^ + dp^; whence 

s= I Vp2 (^6I2 + (^p2 or s = . / \/p2 ^612 + dp\ (1) 

according as ds is expressed in terms of or of p. 

If the formulas of the preceding Article for lengths of 
curves be transformed to polar coordinates by making 
X = p cos $ and y = psin 6, (1) of this Article results. 



240 INTEGRAL CALCULUS 

EXERCISE XXIX. 

1. (a) Find s of the circle p = 2 a cos d, and tlie circumference. 

s = C V4a2cos20 + 4a2sin2 dB = 2a C dd 

= 2a(0-0o)T = 27ra. 
Jo 

(6) For the circle p = a; 

fQ pB ~127r 

s = I pdB = a \ dQ = a{Q -B^)\ = 27ra. 

JBa JOo Jo 

2. Find s of the cardioid p = 2 a (1 — cos d), and total length. 
Here dp = 2 a sin dd; 

:. s = f^ [4 a2 (1 - cos oy + ia^ sin^ d]^ dd 

J do 

= 2a C V2 {1 - cosd)' dd '= 4:a f sin-dd 
JOo Jdo 2 

= 8 a cos ^ — cos 2 = 16 a. 

3. Find s of the spiral of Archimedes p = cud, and the length of the 
first spire. Here 

ds = Va202 + a2 dd = a VT^^dB] 

... s = a f Vl+T^ dd = ^\d Vi + 02 + log (0 + vTT^)T'''' 

= a [tt Vl +47r2 + i log (27r + Vl+47r2)]. 

4. Find s of the logarithmic spiral p = e°^ from po to p, and the length 
from the pole (p = 0) to p = 1. 

aB = logp; .'. add = — or pdd = — ; 
p a 



s 



5. Find s of the conchoid p = a sec d, and the length of the arc from 
d = Oiod = ir/4. Here 

dp = asec0tan0d0; 

... s= ( Va2 sec2 + a2 sec^ d tan^ ddd = a C sec^Bdd 

Jdo Jdo 



a tan d\ = a (tan — tan do) = a. 
J»o J 



CURVE OF A CORD 



241 



144. Curve of a Cord under Uniform Horizontal Load — 
Parabola. — When a flexible cord supports a load which is 
uniformly distributed over the horizontal projection of the 
cord, as, for example, the cable of a suspension bridge, 
which supports a load distributed uniformly per foot of 
roadway, the curve assumed 
by the cord is a parabola. 
This is evident geometri- 
cally, and the curve can be 
shown analytically to be a 
parabola. 

Consider a portion OP of 
the cord AOB, being the 
lowest point of the cord. 
The equihbrating forces 
acting on OP are the ten- 
sions H and T at the ends, 

and the weight of the load, acting at the center of OM, 
Since three forces to be in equilibrium must meet in a point, 
the tangent to the cord at P passes through the middle of 
OM. Now it is a property of the parabola that the sub- 
tangent is bisected at the vertex, in which case OC = ^ 
NP = CM. Hence the curve assumed by the cord is a 
parabola. 

Otherwise the analytic conditions of equiUbrium give 




XMp* = 



WX':^- Hy 



0; 



y- 2H^' 



(1) 



the equation of a parabola. The same result is gotten by 
taking the sides of ^he triangle PDT to represent the three 
forces T, H, and wx, and then, 



dy 
dx 



wx . _ A wx dx _ wo^ 
~h'' '' ^~ Jo ~H~~ 2H 



(1) 



* Algebraic sum of moments of forces about point P. 



242 INTEGRAL CALCULUS 

T coscj) = H and T sin = wx give by division 

, '■ wx dy , „ 

tan ^ — ~Tj = -f^ as before. 

Also, T = = Vi/2 _|_ {y)xY gives the tension at any 

point of the cord. 

The curve is thus shown to be a parabola with its vertex 
at and its axis vertical. If the supports at A and B are at 
the same elevation and I is the span, the sag d at is y, for 

X ^^ 2 l) 

.-. H = f.', and Ti = -^. (3) 

8 a cos 01 

dy ^ , wl 4:d ... 



where <^i is the angle of inclination of PT at the supports. 
The length of the cord for a given span and sag is gotten by 

^dyV'\h , , dy wx , ^„ 

y) dx, where ~f- = -tt and s = OB. 



"CM 

Let V? = -, or — = a, to simplify: then, 
Ha w 

s= f'\i-^t'fdx = - r (a^ + x')i dx (5) 

Jo \_ Cl J ciJo 

W 1 ~|a;=§^ 

Replacing 77 = - , s = Si 

-sv/f^+f„'«^[(-wi^)ai" <« 



CURVE OF A CORD 243 

Total length = 2 OB = 2 si 



wl 
2H 



Corollary. — Expanding the two terms of (7) into series 
(as shown later) and adding Hke terms, the total length of 
the parabola in terms of I, w, and H is 

Total length = Z + 24^ -^^^ + ^jgg^ , (8) 

or in terms of I and d, since B. = ^-^^from (3), 
Total length = Z + _ - -^ + -y^ - • • • . (9) 

Also, expanding (5) by the binomial theorem and integrating 
gives 

a Jo 

aJo L 2a Sa^ 16a^ J 

_ir . x^ x^ x' 1 

~ar^"*"6a 40a3"^ 112^6 ' * J 

, x^ x^ , x"^ , 1 w; 

= ^+6^"40¥^+ll2^ ' ^^"^" a = g- 

.*. Total length, /S = 2 si 

"^24^2 640 JY^"^ 7168 ^6 * * * * ^^ 

The total length can be found by (8) or (9) to any desired 
degree of accuracy and it can be gotten exactly by (7) ; when 
d is quite small compared with I, then the third and succeed- 
ing terms in (8) and (9) are so small that they may be neg- 
lected giving: 

wH^ 8 (P 

Total length, S = l-{- kt-tt^ = ^ + "^Tj approximately. (10) 



244 



INTEGRAL CALCULUS 



When >S and H are given to find I, a cubic equation results, 
the solving of which may be avoided by putting S^ for P, 
since they are nearly equal when c? is small; then, 

2^2^ approximately. (11) 



l = S 



145. The Suspension Bridge. — The cables of a suspen- 
sion bridge are loaded approximately uniformly horizontally, 
since the roadway is horizontal, or nearly so; and the extra 
weight of the cables and the hangers near the supports is a 




small part of the total load carried by the cables. As shown 
in Art. 144, under the conditions stated, the curve of the 

cables is the parabola whose equation \^ y = frjj, where H 

is the horizontal tension. 

Example 1. — The Brooklyn Bridge is a suspension bridge 
which has stays and stiffening trusses to prevent oscillation. 
The span between main towers is 1595 feet. 

If the sag OD is 128 feet and the weight per foot supported 
by each cable is 1200 lbs., without considering the stays or 
stiffening trusses, to find the terminal and horizontal ten- 
sions, substitute the numerical values in (3) of Art. 144. For 
terminal tension, 



Ti = H sec <^i or 
For horizontal tension, 

Ti = /7sec0i = 



-.=v/(f 



+ HK 



2,981,279 lbs 
vol 



Sd 



VZ2 + 16^2 = 3, 128,929 lbs. 



CURVE OF A FLEXIBLE CORD — CATENARY 245 

Example 2. — A cord loaded uniformly horizontally with 
1 lb. per foot is suspended from two supports at same eleva- 
tion and 200 feet apart with a sag at middle of 50 feet. Find 
the length of the cord. 

Here tan (j)i is equal to -|§ = 1 ; 

^ ^ wl 200 w , 
,. tanc/„ = 2^ = ^^=l; 

••• ^ = 260 -• ^=100 lbs. 

= J^ 2 = _J_ 2 

^ 2H^ 200^* 
13 y> 

S = total length = I + 24^2 " 540^4 + * * * 

= 2^^+24f5&- 6S?+ • • • =228.33, approx. 



= V(IooF+(Ioo? + iooiog[l5^±^fiioo)!] 

= 100 V2 + 100 log [1 + V2] = 141.42 + 88.14 
= 229.56, exactly. 

146. Curve of a Flexible Cord — Catenary. — If a per- 
fectly flexible inextensible cord of uniform density and cross 
section be suspended from two fixed points A and 5, it 
assimies a position of equilibrium under the action of gravity. 
The curve thus formed is the Catenary, whether the hne 
joining the points of support is horizontal or not. 

To find the equation of the curve AOB; let w denote the 
weight of a unit length of the cord and s the length of the 
arc whose ends are the lowest point (0, 0) and the point 
(x, y), or P; then ws, the weight of the arc OP, is in equi- 
librium with the tangential tensions at and P. Denote 



246 



INTEGRAL CALCULUS 



the horizontal tension, which is the same at all points, by 
H^wo,. If c ' PT represents the tangential tension T at P, 
c • PD and C' DT will represent, respectively, the horizontal 
and vertical tension at P. 




Hence, 



dy _ c DT _'ws^ _s 
dx~ C' PD wa a 



s _ Vds^ — dx^ 
a 



dx ds 






X ^ p ds ^ ^^ V s + V g' + sn 
a~ Jo Va2 + s2 ^^ L ct J 



The exponential equation is 



e" = 



a 



ae^ 



= s + Va2 + s2, 



which solved for s gives for length of OP, 

s = -(6« — e '^)=a smh - • 
2 \ / a 

(Compare Ex. 5, Art. 142.) 



(1) 



(2) 



(3) 



CURVE OF A FLEXIBLE CORD — CATENARY 247 

du 
Substituting in (3), s = a^ from (1), gives 

g = l(J-e-^)=sinh^; (4) 

y -\- a = -^[e'' -\- e « j = a cosh- (5) 

is the equation of the catenary referred to the axes OX, OY. 
If the origin of coordinates is taken at a distance a below 
the lowest point of the curve and the curve be referred to 
the axes OiXi and OiY, its equation is 

(X x\ 

The horizontal line through Oi is called the directrix of the 
catenary, and Oi is called the origin. 

Corollary I. — Since a = OiO is the length of the cord 
whose weight is equal to the horizontal tension, and there- 
fore the tension at the lowest point 0, it follows that if 
the part AO of the curve were removed and a cord of length 
a, and of the same weight per unit length as the cord of 
the curve, were joined to the arc OP and suspended over a 
smooth peg at 0, the curve OPB would still be in equilibrium. 

Corollary II. — Since the sides of the triangle PTD are 
proportional and parallel to the three forces under which 
the arc OP is in equiHbrium, it follows that: 



tension at P cPT 


T^ 


ds y 


tension at c • PD 


wa 


dx a' 



from (3) and (5i), by differentiating (3); 

/. T = wy; 
that is, the tension at any point of the catenary is equal to the 



248 INTEGRAL CALCULUS 

weight of a portion of the cord whose length is equal to the ordi- 
nate at that point. 

Therefore, if a cord of uniform density and cross section 
hangs freely over any two smooth pegs, the vertical portions 
which hang over the pegs must each terminate on the direc- 
trix of the catenary. 

Corollary III. — Subtracting the square of (3) from the 
square of (5i) gives 

y' = s' + a'; (6) 

dx^ 
and from (6), after substituting a^ = V^jj fi"om Corollary II, 

s^yf, (7) 

From M, the foot of the ordinate at P, draw the perpendicu- 
lar MTi, then PTx = y cos MPTi = Vj-y which in (7) gives 

PT^ = s = the arc OP; (8) 

and since y' = PTi' + TiM% from (6) and (8), 

TiM = a. (9) 

Therefore, the point Ti is on the involute of the catenary 
which originates from the curve at 0; TiM is a tangent to 
this involute; and TiP, the tangent to the catenary, is 
normal to the involute. As TiM is the tangent to this last 
curve, and is equal to he constant quantity a, the involute 
is the equitangential curve, or tractrix. 

147. Expansion of cosh x/a and sinh x/a. — Expanding 

X X 

e^ and e ^^ in series and taking the sum of the two series term 
by term gives for (5i), 



j/ = acosh^ = a[l + ^ + ^ + 



/y.2 /V.4 



APPROXIMATE FORMULAS 249 

Taking the difference of the two series gives for (3), 

s = a sinh - = a - + -^-5-: + -r-^-, + • • • 
a \_a a^Sl a^5\ J 

" ^ "^ 6^ "^ 120^4 + • • • ; (11) 

/. for (4), 

X 

For these expansions, put - for x in Examples 8 and 7 of 

Exercise XLIII. 

148. Approximate Formulas. — The series in Art. 147 
are rapidly convergent when x is small compared with a. 

X 

Near the origin - is a small quantity, s is nearly the same as 

dxi s X 
X, and -y- = - - -\ hence, neglecting the terms with the 
(XX CL a 

higher powers of x; 

(x^ \ x^ X? 

^+^!J = " + 2^' "^ 2/-a = 2^; (1) 

(X X^ \ x^ 



s 



^ = tan0 = - + -|^ = - + ~ (3) 

Equation (1) is the equation of a parabola with its vertex 

at (0, a), the lowest point of the catenary. Where the cord 

is very taut with small sag, s is very nearly the same as x, 

and as they are nearly equal in any case near the vertex 

dii 
where -p is small, if x is put for s in (1) of Art. 146, then 

dy _ ^ . . _ n xdx _ x^ 
dx~ a^ " Jo a 2a' 

the equation of the parabola with its vertex at (0,0) or 0. 



250 INTEGRAL CALCULUS 

Hence, near its lowest point the catenary approximates 
in shape a parabola. When xi = ^l,yi = a-\'d,d denoting 
the sag; hence, for the sag at the lowest point of the curve, 

or, approximately, 
d = ^ = ||; ••• i? = g [(2)ancl(3),Art. 144]. (4) 



Also at the supports tan </>! is, approximately 
2a~2H~l 



tan0i = ;^ = ^ = i^ [(4), Art. 144]. (5) 



When ^1 = 2' 

^'"2"^48a2+ ' ' ' "2 + 48:^2+" ' "2 + "6l"''* " ' ^^^ 

Hence, when the supports are at the same elevation, 

73 ..,273 

Total length = 2si = Z + ^rf^ = ? -f 



24a2 ' 24/^2 

= Z + ^, approx. [(10), Art. 144]. (7) 

Note* — When the sag is 1 per cent of the span, the error 
in H or d, from using these approximate formulas, compared 
with the value from those for the true catenary, is about 2^4 
of 1 per cent. For a sag of 10 per cent of the span, the error 
is about 2 per cent. 

149. Solution of s = a sinh x/a. — While in practice the 
approximate parabolic formulas of the preceding article are 

generally used, the curve y = a cosh - appropriately mag- 

CL 

nified fits any cord hanging under its own weight, the con- 
stant a depending upon the tautness of the cord. 

When the horizontal distance Xi, from the lowest point of 

* Poorman's Applied Mechanics. 



SOLUTION FOR THE CATENARY 251 

the curve to one of the supports B, and the length of the arc 
OB = si are given, then approximate values of a, yi, Ti, H, 
and d = yi — a can be found. For putting Xi for x in (3), 
then 



prVe^— e "y = asinh— ) 



Si = ■p:\e'^ — e "/ = asinh— J (1) 

2 a 

where Si and Xi being given, a is found by solving the trans- 
cendental equation. The solution is by approximations and 
use of tables of the hyperbolic functions. When the supports 
are at the same elevation and I denotes their horizontal 
distance apart or the span, I = 2 Xi, and for total length of 
cord, 

S = 2si = a\e^- e"^^), (2) 

where S and I being given, a is found by solving the equation. 
The curve heretofore considered is called the Common 
Catenary, the cord being of uniform cross section. 

In the Catenary of Uniform Strength, the area of the cross 
section of the cord at any point is varied so there is a con- 
stant tension per unit area of cross section. The equation 
is 

y = clog sec (x/c). 

Example 1. — A chain 62 feet long, weighing 20 lbs. per 
ft., is suspended at two points in a horizontal line 50 ft. apart. 
Find a; the horizontal tension; the terminal tension; the 
sag of the chain. 

Here 

s = ^\e^-e «; = ^Ve«-e «; = 31. (1) 



a: + g^ = 25 + 1^ = 31, by (2), Art. 148; 



^ _ o. . (25) 

(25)^ ^ 6X6 . 
a2 25 ' 

— = "E ~ J- '2, approximately. 
a o 



252 INTEGRAL CALCULUS 

25 25 

Now let — = z: ^, a = — , which substituted in (1) gives 
a z 

Let j{z) = e" — e~^ — 2.48 2: = 0, where 1.2 is approx- 
imate value for z. Reference to a table of hyperbolic sines 
will show that the value of z is between 1.1 and 1.2. 

The following is a method of getting a closer approxima- 
tion to the value of the root z without the use of tables. Let 
z = Zi-\-h, where Zi is an approximate root differing from 
the root by a small quantity h. 

By Taylor's Theorem: 

J{z) = /fe + h) =f{z,) + hf fe) + |V"fe) + • • • = 0. 
Neglecting higher powers of h,f{zi) + hf (zi) = 0; 

.', z = Zi — n,^ Y as first approximation for z. 

Let this value be Z2, differing from zhyk < h; 

1.2 
.-. f{z) =f{z, + k) =f{z.2) + hf fe) + |/"fe) + • • • = 0. 

Neglecting higher powers of k, f (Zi) + hf (22) = 0; 

•• " /'fe)' •• ' ^' f'izd 

as second approximate value for z. 

By repeating this process a value which approximates 
closer and closer to the true value of the root can be gotten. 
Applying the method to this example: 

fiz,) _ e''^-e-^-'-2ASXl.2 _ 0.04 _ ,,,, 
f(^i)~ ei-2 + 6-1-2 -2.48 - 1.14- "-^^^^ 

/. Z2 = Zi-h = 1.2- 0.035 = 1.165. 



SOLUTION FOR THE CATENARY 253 

Now 



"' /,1.165 _L_ x,-1.165 O /1C 1 rkOQ V^.WUU, 



giies - g -Li65 _ 2.48 X 1.165 ^ _ 0.005 
4_ e-1.165 _ 2.48 ~ 1.038 
/. 2 = 2:2 - A; = 1.165 - 0.005 = 1.16, 

an approximation close enough, the value found from the 
table by interpolation being the same. 
Hence, 

a = ?5 = ^ = 21.55 ft. 
z 1.16 

f^ = tya = 20 X 21.55 = 431 lbs. 

2/1 = Vsi2 + a2= V(31)2 + (21.55)^^ = 37.75 ft. 

(6) Cot. Ill, Art. 146. 
J = 2/1 - a = 37.75 - 21.55 = 16.20 ft. 
r = ^2/1 = 20 X 37.75 = 755 lbs. Cor. 11, Art. 146. 

Example 2. — To get the correction for the sag in measur- 
ing horizontal distances with a steel tape unsupported except 
at the ends where puU is apphed. 

' A 100 ft. steel tape was weighed and its weight found to 
be If lbs., making w = 0.0175 lb. 

Using the equation of the parabola, 

y = ^-p, where P = H is the pull in pounds; 

d = ^^, exact if I is span, approx. if I is arc or tape, 
o Jr 

Length of tape = s = 1 + -^y — -^-^ + • • • 

= ^ + 24P"2 ~ 640P"4 + • • • ^ (8), Art. 144; 
_ _7_8c^2_ ^2^3 C approx. correction for one ( .. 
^~^ ^^~24P2 ^tapelength, (10),Art. 144^' ^^ 
True horizontal distance 

ID 9i 

= Z = s — . p^ , approx., s^ being put for P. (11), Art. 144. (2) 



254 INTEGRAL CALCULUS 

In (1) when s and P are given to find I a cubic equation 
results, the solving of which may be avoided by putting s^ 
for l^, since they are nearly equal, the correction being small. 
Substituting the pull P in lbs. in (1) or (2), the correction or 
the horizontal distance, respectively, for each tape length 
will be found. The pull may be measured with a spring 
balance. Since sag shortens the distance between tape ends 
and pull lengthens it by stretching the steel, there is some 
pull at which the two effects are balanced. The stretch is 
given by 

where P is pull in lbs., I is length of tape, A is area of the 
cross section, and E is modulus of elasticity, usually expressed 
in lbs. per sq. in.; for steel, E = 30,000,000 lbs. per sq. in. 

To find the pull that will just balance the effect of sag, 
put the values of x from (1) and y from (3) equal; 

wH' PI . ^ , 

= -rr=, approximately; 



24 P2 AE 
whence, 



-^- 



-^ — , approximately. 



In this example the width of the tape was x% in. and the 
thickness ^V in., making area A = 0.00525 sq. in. 



V 2^ 



'^^^'^•^^(100)^30 = 27 lbs. 



24 (1000) 

This pull to balance effect of sag can be determined experi- 
mentally by marking on a horizontal surface two points 100 
feet apart, and then noting the pull on a spring balance when 
the ends of the suspended tape are exactly over the points 
marked. 

150. The Tractrix. — The characteristic property of the 
tractrix is that the length of its tangent at any point is 



THE TRACTRIX 255 

constant. Denote the constant length of the tangent PT 
by a. 

If a string of length a has a weight attached at one end 
while the other end moves along OX in a rough horizontal 
plane XOY^ the point P of the weight, as it is drawn over 
the plane, will trace the tractrix APPi .... 



Let AO be the initial position of the string and PT any 
intermediate position. Since at every instant the force 
exerted on the weight at P is in the direction of the string 
PT, the motion of the point P must be in the same direction; 
that is, the direction of the tractrix at P is the same as that 
of the line PT, which is, therefore, a tangent to the curve. 

To find the length and equation of the curve: let 

PD = ds; then -PN = dy and ND = dx; 

ds^ _ _ Pp^ _ _ ^ (^\ 

•*• dy- PN~ y ^^^ 

Hence, if s is reckoned from A (0, a), then the length 

s=-a p* = alog^. (2) 

Ja y y 

Also, from the figure, 

^ = - ^ or — = - ^^' ~ ^' > (3) 

dx Va2 - 2/2' dy V ' 



256 INTEGRAL CALCULUS 

/. x= r - Va^-y^^= - Va2 - y^ 

+ a\og\^ — ^—^j (See Ex. 20, Exercise XXV.) 

is the equation of the tractrix. 

Example 1. — To find the area bounded by the tractrix in 
the first quadrant, the x-axis, and the y-axis. 

A = lydx=- fVa^-y^dy (from (3)) • 

«>'0 *Ja 

Example 2. — To find the area bounded by the catenary, 
the X-axis, the y-axis, and any ordinate y. 

A=^ f\e^ + e" V dx = ^- (e« - e~«) = a^ sinh - (1) 
Z t/ Z a 

= a ^ Ve"^ — e "/ = as, 

where s = length of arc. (See (3), Art. 146.) 
For area up to x = a. 

Hence, by (1), 

a cosh -dx = a^ sinh - • 
a a 

151. Evolute of the Tractrix. — It has been shown in 
Art. 146, Corollary III, that the involute of the catenary is the 
tractrix. Conversely, it may be shown that the evolute of 
the tractrix is the catenary. Let {a, (3) be the coordinates of 
C, the point of intersection of the normal at any point 
P (x, y) of the tractrix and the perpendicular to x-axis at T, 
end of tangent, PT = a. From the figure (Art. 150), 

or = « = X + Va2 - y\ (1) 



EVOLUTE OF THE TRACTRIX 257 



and {TC and PC being drawn) 
TC = jS = -, since 

y 

Equation of tractrix, 



^^ ^ a^ . /3 a ( from similar triangles , .„. 
TC = ^ = -, smce - = -\ ^^^ a iDrri^j V (2) 
y a y \ PTC and PTM ' 



(3) 



x= - V a2 - 1/2 + a log — ^ 

Eliminating x and y from these three equations gives 

a L a J 

.-. e^ = ^-±^^!ZZ. (4) 

Solving (4) for /3 gives for the relation between a and j8, 

which is the equation of the catenary with origin at 0; and 
its lowest point at A, one end of the tractrix. 

The normal PC to the tractrix, the involute, is the tangent 
to the catenary, the evolute, and is equal in length to the 
arc AC (not drawn) of the catenary. 

If the equations for coordinates, a and jS, of the center of 
curvature, given in Art. 94, be used, the values found in 
(1) and (2) will result. 

Note. — It may be shown independently of Art. 146, 
Corollary III, that the catenary is the curve which has the 
property that the line drawn from the foot of any ordinate 
of the curve perpendicular to the corresponding tangent is 
of constant length a. 

Thus, let B be the angle which the tangent CP makes with the 
jr-axis and it is evident from the figure {CP being drawn) that 
a . 1 1 

COS0 = 



^ Vl+tan2^ 



Mt) 



258 INTEGRAL CALCULUS 



da 



a v^ 



(I Jo Ja 



d^ 



.: ^ = log(, + V,-^)T=log(^±^^); 

U Ja \ G / 

... ^ = l±^^FEI. (4) 

a 

a( - --\ 
Hence, j8 = ^ V^'" + ^ "A as before. 



CHAPTER IV. 

INTEGRATION AS THE LIMIT OF A SUM. SUR- 
FACES AND VOLUMES. 

152. Limit of a Sum. — A definite integral has been 
defined (Art. 124) as an increment of an indefinite integral. 

It will now be shown that a definite integral equals the limit 
of the sum of an infinite number of infinitesimal increments or 
differentials. 

Many problems in pm'e and applied mathematics can be 
brought under the following form : 

Given a continuous function, y = f (x), from x = a to 
X = h. Divide the interval from x = a to x = b into n equal 
parts, of length Ao^ = (6 — a)/n. Let Xi, X2, X3, . . . Xn be 
values of x, one in each interval; take the value of the function 
at each of these points, and multiply by Ax; then form the sum: 
/(xi)Ax+/(x2)Ax+ . • . +/(xn)Ax. (1) 

Required, the limit of this sum, as n increases indefinitely and 
Ax approaches zero. 

This problem may be interpreted geometrically as the 
problem of finding the area under the curve y = f (x)j 
between the ordinates x = a and x = b; each term of the 
sum representing the area of a rectangle whose base is Ax 
and whose altitude is the height of the curve at one of the 
points selected. 

Let the area MiPiBX be denoted by A; let OMi = a, 
OX = b, and PiB be the locus oi y = f (x). 

Let Ax be one of the equal parts of MiX, although the 
parts need not be made equal provided the largest of them 
approaches zero when n is made to increase indefinitely. 

259 



260 



INTEGRAL CALCULUS 



It is easily seen that the difference between the sum of the 
rectangles as formed and the area A is less than a rectangle 
whose base is Ax and whose altitude is a constant, / (6) — 
/ (a). Since this difference approaches zero as Ax = 0. the 
sum of either set of rectangles approaches the area A as a 



M^ M^ M^ M^ M^ M^X 

limit. It is evident that the sum of the rectangles which 
are partly above the curve is greater than A, while the sum 
of those which are wholly under the curve is less than A. 
By the notation of a sum, letting T be the difference, 

A = ^ j{x)Ax±.T, where T, <f{h)Ax, =0, as Ax = 0; 

a 

:. limit T V (^) Ao; = A = T / (x) dx. (2) 

Ax=0 a *J a 

The equation (2) is true, for it has been already shown that 
A = Jj (x) dx = Jf (x) dx~^^^- ff (x) dx]^^ = F(b)-F (a), 
where j f(x) dx = F (x). It follows that the hmit of (1) is 
\im\f(x,)Ax+f{x2)Ax+- ' '+f(xn)Ax] = F{h)-F(a), (3) 
where a and b are end values of x and / f(x)dx = F(x), 



THE SUMMATION PROCESS 



261 




Tra' 



The theorem of this Article summarized in equation (2) may 
be said to be the fundamental theorem 
of the Integral Calculus. 

As a simple example of the determi- 
nation of an area by getting the limit 
of a sum of an indefinite number of in- 
finitesimal elements of area, let a circle 
of radius a be divided into concentric 
rings of width Ax; then for the area 
A, 

A = limit y 27ra;Ax = 27r / xdx = 2-^x^12 = 

Here AA = 2 tt (a; + i Ax) Aa; and dA = 2t^x dx. 

153. The Summation Process. — On account of the 
frequency of the occurrence of the summation process, it 
may be said that an integral means the limit of a sum, the 
Hmit being in most cases most easily found as an anti- 
differential or anti-derivative ; that is, by the inverse process 
to differentiation, namely, by integration. 

The symbol / for integration, the elongated S, is derived 

from the initial letter of summa, the integral being originally 
conceived as a definite integral, the limit of a sum. Accord- 
ing to Art. 128, the indefinite integral also may be regarded 
as the limit of a sum. 

The fact that the summation of an indefinitely large 
number of indefinitely small terms is in most cases easily 
effected by a comparatively simple process is of the highest 
importance. Thus integration replaces the tedious and often 
difficult process of direct summation and gives an exact 
result, while the other often gives but an approximation at 
the best. 

While the process of summation has been illustrated 
geometrically by the determination of an area, the reason 



262 INTEGRAL CALCULUS 

of the process by no means depends upon geometrical con- 
siderations. The method is appHcable to the determination 
of the hmit of the sum of small magnitudes of all kinds — 
volumes, masses, velocities, pressures, heat, work, etc. For 
an example of finding the hmit of the sum of small volumes, 
consider the volume V generated by revolving the area 
MiPiBX of the figure of Art. 152 about OX as an axis. 
Each of the rectangles P1M2, . . • , PnX will generate a 
cylinder whose volume will be expressed by t {f {x)y Ax; 
hence, 

a 

where T, < tt (/ {h)y Ax, = 0, as Ax = 0; 

/. hmit y.^TT (/ {x)Y Ax=V = Ttt (/ (x))2 dx, (1) 

Example. — Find volume of a sphere by revolution of 

y"^ = o? — x^ \V = I IT (a^ — x^) dx = ^ ira^. 

For another example of finding the limit of the sum of 
small volumes, find the volume of the sphere considered as 
made up of concentric shells of thickness Ap. 

F = lim T4 7rp2 . Ap = 4 TT / p2 dp = | wa^. 

154. Approximate and Exact Summations. — When the 
rate of change (or the derivative) of a variable quantity is 
given, the total amount (or the integral of the rate) can be 
obtained approximately by direct summation, and exactly 
by finding the limit of a sum; that is, by integration. 

For example, suppose the speed of a train is increasing 
uniformly from zero to 60 miles per hour, in 88 seconds; 
that is, from zero to 88 ft. per sec. in 88 seconds, the increase 
in speed each second (the acceleration) is 1 foot per second. 



APPROXIMATE AND EXACT SUMMATIONS 263 

Hence the speeds at the beginnings of each of the seconds 
are 0, 1, 2, 3, ... , etc. 

Taking the speeds as approximately the same during each 
second as at the beginnings, the total distance. 

07,00 
s = + l+2 + 3+ • • • +86+87 = ^^ = 3828 ft., 

which is evidently less than the true distance. 

Taking the speed at the end of a second as that during 

the second, 

00. oq 
s =1+2 + 3+4+ . • • +87 + 88 = ^Y^ = 3916ft., 

which is evidently greater than the true distance. These 
values for the distance differ by 88 ft. and it is certain that 
the true distance is between 3828 ft. and 3916 ft. When the 
length of the interval during which the speed is taken as 
constant is reduced more and more, the result will be more 
and more accurate, nearer and nearer to the true distance. 
Manifestly, the exact distance is the limit approached by 
this simimation of small distances as the interval of time 
Lt approaches zero: 

]<=88 '=88 /»<=88 /2-l<=88 

= \imy.vM= I tdt = }.\ = 3872ft. 

In general, 

s = limit ^ V At = I atdt = i at^, 

a being constant acceleration. 

In mechanics, the determinations of centers of gravity, 
centers of pressure, moments of inertia, varying stress, etc., 
involve the summation principle; and the greater number 
of the integrations in practice appear more naturally as 
limits of sums than as reversed rates, anti-derivatives, or 
anti-differentials. 

The summation of an infinite number of terms is always 



264 INTEGRAL CALCULUS 

involved when one of the factors entering into the problem 
varies continuously. For example, in the problem of finding 
the mass of a body, defined as the product of density and 
volume; when the density p varies continuously, 



m 



limit ypA7= fpdV, 

A7=0 J 



where the integral taken between 'limits," that is, with 
end values for the independent variable, is the limit required. 
Hence the mass is given by a definite integral, which can be 
evaluated when the density p is a known function of the 
volume V, that is, of the variables x, y, z or r, d, cj), in terms 
of which the volume may be expressed. When the density 
p is constant, it is evident that the mass is 



m 



= 5)pAy = p fdV = pV. 



Thus, when the body is composed of different liquids of 
varying densities in the layers or strata, the total mass is 
found by the addition of a finite number of terms. For if 
Vi, V2, V3, . . . Vn denote the volumes of the separate 
parts, and pi, p2, ps, . . • Pn the corresponding densities, then 

^ = plVi + P2V2 + P3V3 + • • • + PnVn, 

where the summation is made without integration. 

The above will give an approximate result even when the 
density varies throughout the whole mass. When, however, 
the density varies continuously as in the atmosphere, the 
total volume is divided into n parts each equal to AF and 
each part is multiplied by the density at that part of the 
body. There are then n elements of the form pAF, and 
when n is finite their summation will be an approximation 
to the mass of the whole; but to get the exact value, the 
limit of the sum, as n becomes infinite and AF = 0, must be 
found, and hence the exact value of the whole mass is deter- 
mined by the process of integration. 



APPROXIMATE AND EXACT SUMMATIONS 265 



Example 1. — li y = x^, find ^ x^ Ax for different values 

,^2 1 

of Ax, and get lim ^ x^ Ax. Get lim V x"^ Ax. 



Ax=0 '1 

When Ax = 0.2, 

,2 



Ax=0 



^ x^ Ax = (V + l72 -\-h? + i:S' + Is') 0.2 = 2.04. 

When Ax = 0.1, 

2)%2^x= (p + n' + r2'+ • • • + 19') 0.1 = 2.18. 

When Aa; = 0.05, 

V%2 A:c = (P + iM' + n' + • • . + L9?) 0.05 = 2.26. 

X-N^ /*2 ^3"]2 8—1 

Lim 2^ x'^ Ax = I x^dx = -^\ = — - — 

Ax=0 ^1 J I oji d 

= 2. 33 J square units in M1P1P2M2. 

SI ri ^3-11 I 

x"^ Ax = I x^dx = ^\ = - 
Jo / ojo o 

= 0.334 = J of rectangle OikTiPiiVi. 



K 



I 
/ 





^, 



^o 



1 ^4^^ 

Example 2. — If y = -, find ^ — for different values of 

X ^^ 1 X 



.4 Ax 



Ax 



Ax, and get hm V — Get V — as Ax = 0, 

Ax=0^1 0? ^0 X 



266 INTEGRAL CALCULUS 

When Ax = 1, 

When ^x = 0.5, T — = 1.593. 

'^1 X 

— = 1.426. 

1 X 

LimT-^= / -:^ = log:c = log4- logl = log4 

Ax=0 ^1 X Ji X Ji 

= 1.386 = Area M1P1P4M4. 
LimT — = I — = loga; =logl-loga= — loga =00; 

Ax=0 a ^ t/a ^ Ja Ja=0 

hence, when a = 0, the Hmit does not exist, as T, — = qo . 

^Q X Jax=0 

(Compare Ex. 11, Art. 135.) 
Note. — For examples of appHcation see Art. 189. 

EXERCISE XXX. 

1. If ?/ = X, find ^ a: Ax, when Ax = 1; when Ax = 0.5; when 

Ax=»0.2. Getlim V xAx. Ans. 18; 19; 19.6. 

Ax=o '^^a 

Ans. 20. 

2. If 2/ = tan 0, find ^ tan A0, when A0 = ^ ; when A0 = ^ ; 

I 

2E 

whenA0 = r^. Get Hm V'tan^A^. Ans. 0.316; 0.328; 0.340. 

15U Ae=o ^^-^ /- 

^ Ans. loge V2 = 0.346. 

Determine the following quantities (a) approximately by summation 

of a limited number of terms; (b) exactly by finding the limit of the'sum 

of an infinite number of terms by integration. 

3. The area under the curve y = a?, from x = to x = 2; from 
a; «= —1 to X = 1. 

4. The distance passed over by a body falling with constant accelera- 
tion g = 32.2 per sec.^, from t =^ 1 to t = 4, v = gt being the relation 
of V and t. 



VOLUMES 



267 



6. The increase in speed of a body falling with acceleration of 
g = 32.2 per sec.2, from t = Otot = 3. 

6. The number of revolutions made in 5 minutes by a wheel which 
revolves with angular speed co = ^VlOOO radians per second. 

7. The time required by the wheel of Ex. 6 to make the first ten 
revolutions. 

155. Volumes. — The volumes of most solids may be 
found approximately by the summation of a finite number of 
parts and exactly by finding the limit of the sum of an infinite 
number of terms by integration. 




Example. — To find the volimie of the right circular cone 
whose altitude is h and the radius of whose base is a. Divid- 
ing the volume into parts, each A 7, by passing planes Ax 
apart parallel to the base Ah, and denoting a section at a 
distance x from the vertex at the origin by Ax, then, since 
Ax/ Ah = x^/h^, V is given approximately by 



XA7 = T AxAx = X 
and exactly by 



Ah ^ Ax 



7 = hm X ^k 

Ax=0 

^ Ah x^> 
/i2 3 



h^ 



Ax 



^Ah p 
h' Jo 



x'^dx 



I A./,. 



(1) 



(2) 



(3) 



While AF is a frustum of the cone, dV may be represented 
by the cyhnder PMMi = Ax' Ax = iry^ dx. 

It is to be noted that the equations all apply to a pyramid 
with any plane base Ah^^ well as to the cone. 



268 



INTEGRAL CALCULUS 



For another example : to find the volume of a sphere with 
radius a, divide by planes perpendicular to OX; then, since 

V = lim X' A,^-^^x = ^ r {a' - x^) dx 

Ax=0 —a Ci (X *J — a 

Aof . x^l" Ao 4 „ 4 „ - . „ 

= -r a^a; — 17 = — ? * o ^ = o 7^o^^ where Ao = wa^, 
c? \_ 3J_a a^ 3 3 




Otherwise; 

2) AF = 2) Ax Aaj = 2) ^?/^ ^^' where Ax = tti/^; 

x-\ r« 4 

.*. F = Hm X TTi/^ Ax = TT I (a^ — x^) dx = -^ iroF. 

156. Representation of a Volume by an Area. — In Art. 
138 on the significance of an area as an integral it was stated 
that the integrals represented by areas might be functions 
of various kinds. To show an example of a volume as an 
integral represented by an area under a curve, let the volume 
of the paraboloid of revolution, between x = and x = 4, 
be first found as the limit of the sum of the parts between 



REPRESENTATION OF A VOLUME BY AN AREA 269 

the parallel planes Ax apart, as Ax = and the number 
of the parts increases without limit. The equation of the 
generating parabola being y'^ = I x, 

V = Umit y\ rf Ax = -T I xdx = -r-pr\— 2Tr cubic units. 
Ax=o ^0 - 4 Jo 4 2 Jo 




•^JC 



To represent this volume graphically by an area, the Kne OP^ 

is drawn by the equation y = jx, this being the function 

which was integrated to get the volume of the soUd P^OP^. 

Producing the ordinate M2P2 to P", the area OM2P" 

graphically represents the volume of the sohd P^OP'i- For, 

Area OM^P" = I ydx=j I xdx=j-^\ =2Tr square units. 



270 INTEGRAL CALCULUS 

The last result may be verified by noting that the ordinate 
M2P'\ for X = 4, being t, the area of the triangle is 2 tt. 

In the same way it may be seen that any part of the area, 
as OMP', represents the corresponding part of the volume 
of the soHd; that is, there is the same number of square units 
in the one as there are cubic units in the other. 

If OP'P"'y the first integral curve of OP'P"j whose equa- 
tion is 



=x 



-rxdx = -^ (see Art. 140) 



be drawn, its ordinates will represent both the areas of the 
parts of OMiP'' and the volumes of the parts of the parab- 
oloid measured from ; that is, the measure of the ordinates 
in linear units will be the same as that of the areas in square 
units and that of the volumes in cubic units. 

Length of M2P'" ^ V — ~o~\ = 2 tt Hnear units. 
Note. — The volume of the cone of Art. 155 may be graphi- 
cally represented by the area under the parabola y = -r^x^, 

and the volume of the sphere by the area under the parabola 
2/ = TT (a^ — x^) . If the first integral curves. 

An 



3^,x3 and 



= .(a^x-|) 



be drawn, their ordinates will represent both the areas and the 
volumes in the two cases, respectively. 

157. Surface and Volume of Any Frustum. — A solid 
bounded by two parallel planes is, in general, called a 
frustum. One or both of the truncating planes may in 
special cases, as in the sphere, touch the frustum in only 
one point and be tangent planes. 

The method of dividing the soHd into thin slices and taking 
the sum of the approximate expressions for the small parts 
as an approximate expression for the whole, and taking the 



SURFACE AND VOLUME OF ANY FRUSTUM 271 

limit of the sum as an exact expression for the whole, may be 
applied to any solid even when the solid is not regular and 
the sections not regular plane curves. 




Let the solid represented in the figure be divided into 
slices by planes perpendicular to an axis OX; then, taking 
Ax^x as an approximate expression for the volume of the 
sHce P — NiMiRi, Ax being the area of the section PNMR 
at a distance x from plane ZOY, the approximate expression 
is 

x=h 



X^v = XA.^x, 



where h{= OA) is the distance between the truncating or 
bounding planes. The exact expression is 

V = \imy,Ax^x= I Axdx. (1) 

A„^n-*-< Jx=0 



Ax=0 



x=0 



When the area of a section is a function of the distance x 
from one of the bounding planes and hence Ax can be ex- 
pressed in terms of x, the limit may be found by integration. 
The frustum formula for volumes is, therefore, 



Jr*x = h px=h 

F{x)dx or V= I 
x=0 Jx=hi 



F{x)dx, (10 



272 INTEGRAL CALCULUS 

where Ax is F(x), some function of x; the one form giving 
the whole volume and the other a segment or any part 
thereof. 

To get expressions for the area of the surface S, let P be 
the curve NPR, then AS = NPRRiP'Ni, and the approxi- 
mate expression is 

where As = NNi and s is the length of CN. 
The exact expression for the surface is 

S = ]imity\PAs= I Pds. (2) 

As=0 s=Q Js=0 

When the curve P is a function of s, the bounding curve in 
XZ plane, and can be expressed in terms of s, or when ds 
can be expressed in terms of P, with change of end values, 
the limit can be found by integration. If the surface S is 
conceived as generated by the curve NPR as it moves with 
its plane always perpendicular to OX, when its plane is in 
the position as shown, at a distance x from plane YZ, let 
NN\ be drawn equal to ds but parallel to OX; then since the 
surface is cylindrical, the increase of S, if the increase became 
uniform, is 

dS^P ds, the surface NPRR'P'Ni'] 

Pds. , (2) 

s=0 

If the curve NPR is a circle, as in soHds of revolution, with 
the center at M on the a;-axis, then P = 2 tt^/ and Ax = irifi 
(2) and (1) becoming 

>S = 27r r^^'yds (3) 

Js=0 

Jrx=h 
y^dXy (4) 

x=0 



SURFACE AND VOLUME OF ANY FRUSTUM 273 

where dV = iry^ dx is the volume of the cyhnder generated 
by the area of the circle iry^, as it moves uniformly through 
Ax = dx. 

Note. — In deriving (1) and (4), in the figure, NNi = Ao; = 
dx; while in deriving (2) and (3), ds, the uniform change of 
5 along a tangent to the curve CN at the point N, is drawn 
parallel to OX and represented in length by NNi, although 
it is not the same as Ax = dx but is really longer. 

Example 1. — To find the lateral surface of the cone of 
Art. 155: by (3), 

S = 2^ I yds = 2Trl -^ sds, where s = -y = OPf 

n Si 
-= 2 TT 7 pr = -wal, where I = OPh, an element. 

6 z Jo 

Again, 

J'^Sft /•2/=a I /I \ I 

yds = 2Tr I y-dy, since ds = d(-y]= -dy^ 
Jo ct \a / a 

limit or end value being changed from I to a. 

Example 2. — To find the surface of the paraboloid of 
Art. 156: 

J^l 1 1 

2/ [1 + 64 y^]''dy, from 2/^ = 7 x, 
4 

.[l + (64,^)i]J = ^ ((65)1-1) 



27r 
1281 



0^ (65 V65 — 1) square units. 



274 



INTEGRAL CALCULUS 



Example 3. — To find the surface of the sphere of Art. 155, 
or any part of it, as a zone. 

For a change take origin at A' on the circumference, 
making y = a/2 ax — x^ and 2 iry the curve P bounding the 
section A^; then by (2) or (3), 

Pds = 2t I yds = 2t I adx 

Jq Jxo 

(where y ds = a dx, from similar triangles, OMP and PDT) 

]x I2a 

= 2 7ra (o; — Xo) or 2 i^ax — 4 Tra^. 
xo Jo 



Ddx T 




Drawing PT = ds from P parallel to a;-axis, 2 iry ds is the 
lateral surface of the cylinder PT\ which is equal in area 
to that of the cylinder DT\ which is 2 ira dx. 
The volume is again, with origin at A', by (4),' 



yidx = ir j^ {2ax-x')dx = ir\^-'^\ = 



7ra^ 



Example 4. — To find the lateral surface of a quadrangular 
pyramid. Let Ph = perimeter of base and I = OPh = slant 
height. Let PMN be the position of the generating perim- 
eter P when s = OP. Since P and Ph are similar, 



SURFACE AND VOLUME OF ANY FRUSTUM 275 



P 
Ph 



OP 
OPh 



s 



hence, P = -^ s, in (2) ; 



^ = Jo ^^^ = TJo^^^ = T2jo="2-' 

that is, the convex surface of any pjramid or cone (Ex. 1) 
is measured by half the product of perimeter of base and 
slant height. 
y 




For the volume, 

V= I Axdx = ^i x^dx, since-T^=7^, Ax = areaPilfiV, 

t/o ti Jq Ah ft 

nh I 



Ah X^ 

h^ 3 



that is, the volume of any pyramid or cone ((3), Art. 155) is 
measured by one-third the product of its base and altitude. 

Note. — The foregoing, for the purpose of illustration, have 
been for the most part examples of elementary soUds whose 
surfaces and volumes are known from soUd geometry. The 
fruitfulness of the method is seen in the determination of the 
surfaces and volumes of the frusta of unfamiliar and complex 
solids. 

The following are some examples: 

Example 5. A monument is to be built in horizontal rec- 
tangular sections, one side of a section to vary as the dis- 
tance below the top and the other as the square of this 



276 



INTEGRAL CALCULUS 



distance. The base is to be a square 30 feet on a side, and 
the height of the monument is to be 20 feet. Find the vol- 
ume when it is made up of rectangular blocks with vertical 
sides; and also the volume when the sections vary contin- 
uously from top to base. 

Taking origin at top and z positive downward, let Aa = 
4:xy,2y = az,2x = hz^; 22/ = 20a = 30; /. a -= ^•, 2x = 



400 6 = 30; .*. h 
for curve OA. A, 

2=20 



4^> 



4x2/ 



y = i 

ahz^ = 



z, for line OB; x = -ijjZ^ 



9 ^3 

so 2 ) 



2=20 



1 = 5 . ^9_ [0 + 53 _|_ 103 + 153 + 203] 

Ja2=5 

= 7031i cu. ft. 

Hence, as Az, the thickness 
of the blocks, is made less 
and less, the volume will 
approach 4500 cu. ft., the 
volume when sections vary 
continuously. The plan is 
shown on reduced scale. 
The projection of OD, the 
curve of intersection of the 

JO 




JO 



iB ID 

L .]s J 



\ 


1 


X 






\ ■ 


1 
1 


/ 








/I 


\ 




15 
A 


/ 


1 


\ 




^ 


1 


\ 





IS 



15 



plane surface OBD and the curved surface OAD, is O'D 
on plane parallel to X7 plane. The equation of O'D is 



SURFACE AND VOLUME OF ANY FRUSTUM 277 

if = 15 X, by eliminating z from y = %z and x = ^\ ^. 
The plan shows the corners of the blocks on this curve. 

Example 6. Find the lateral sm-face of the mommient of 
Ex. 5. When built of rectangular blocks, the sum of the 
rectangular areas gives the area of the surface. When the 
stone is shaped to make sections vary continuously, or when 
this is effected by using concrete in shaped forms, find the 
areas of the surfaces OBD and OAT) separately. 

E = OAT>^ fpds = £l.ds ^iP[l +(1)7.. 
3 p Til /3 Vl^^ 3 1600 2r/^ , 9 Ml'^ 

=f[f^!5-']:- 

4 (OAD) = 1|2[46:|^-] ^ 200 ^ 3^ ^ ^ 3^^ sq. ft. 

'=^^^= f''^ = n^''^ = torb + (2)7 '^ 

3 8000 ,^. „^ 
= ^.-^= 125 sq.ft. 

4 (OBD) = 4 X 125 = 500 sq. ft. Total surface = 1362 sq. ft. 

Note. — Since OBD is a plane surface, its area may be 
found by 

A = I xds = I z-^ s^ ds = 777^ • :5- =125 sq. ft., 

Jo Js=0 i^o iZo Ojo 

as above. Here x = yfj s^ is the equation of curve OD in 
the oblique plane of sx, for since OB = 25, y = i s, and 
2/2 = 15 X becomes ^5 s^ = 15 a:, or x = its s^- 

In Ex. 5 above, y'^ = 15 x is given as equation of pro- 
jection of OD on xy plane, or any plane parallel thereto. 



278 INTEGRAL CALCULUS 

While OD is given as a line in space by two equations, by 
rotating axis OY about OX through tan-^ f§ = cos-^ f , it is 
given by one equation, s^ = ^f ^ x, in plane of sx. 

Example 7. Find the volume common to two right circular 
cylinders of equal radius a, whose axes intersect at right 
angles. 



Let the two cylinders be x^ -\- z^ = a^ and y"^ -\- z^ = a^; 
then A, = LMPN = xy = a^ - z^, and 



8 pA. dz = Sr (a2 - ^2) dz = S \ah - ^1' 



16 3 



The total volume common, being 8 times Z — OACB, is ^3^- aP, 
Example 8. A dome has the shape of the figure of Ex. 7, 

find the area of the curved surface. 

The surface ZBC is equal in area to the surface ZAC, and 

is one-eighth part of the surface of the dome, which surface 

is the upper half of the surface of the common volume of 

Ex.7. 

Hence the surface of the dome of eight equal parts is 

given by 



SURFACE AND VOLUME OF ANY FRUSTUM 279 



S = 8ZBC = 8 r^Pds = 8 r^NPds 

-dz = 8a I dz = Sa^. 
y Jo 

The result shows that each of the curved surfaces of the sohd 
Z — OACB is equal in area to 
its base OACB; the surface of 
the dome being just twice that 
of its base. 

Note. — Another determina- ^ 
tion of the area of ZBC may 
be made by developing the 
curved surface upon a plane 
and finding the area as a plane 
area. Thus, developing ZBC 
as the plane area Z'BC, with B as origin; 





area Z^BC = area ZBC 



J\'dz\ 



where a; = x' = a cos 6, and z' = ad 






cos 6d (ad) = a? sin Q 



'I=«- 



Example 9. Given a right cy Under 
of altitude h, and radius of base a. 
Through a diameter of the upper base 
two planes are passed, touching the 

lower base on opposite sides. Find the volume included 

between the planes. 



280 



INTEGRAL CALCULUS 



y = r A, dx = 4 r {MNPR) dx = 4: tyz dx 

t/O t/O Jo 

= 4 r(a''-x')^^ia-x)dx 
Jo a^ ' 

= — pa (a2 - x^)-^ dx-— r (a2 - x^)"^ x dx 
a Jo a Jo 

[_Z Ji ajo o tt Jo 



TTO^h— 5 a2/i= (vol, of cylinder) — (vol. outside the planes). 



Here 



OA ' 
MN = RP = y = (a" - x"^)^. 

lY 



NP = z = -{a-x); 




It may be noted that, when 
h is equal to a, the volume 
outside of the planes being | 
a^, is one-fourth of the volume 
common to the two cyhnders of 
Ex.7. 

Example 10. Two cyhnders 
of equal altitude h have a 
circle of radius a, for their 
common upper base. Their 
lower bases are tangent to each 
other. Find the volume com- 
mon to the two cylinders. 



7= f^Aydy^ r {PMM')dy= txzdy 

*y hi t/ —a J —a 

= I x-xdy = - I x^dy=- I {a^ — y^)dy 



SURFACE AND VOLUME OF ANY FRUSTUM 281 



Here, PMM' being similar to ZAA'j 



NP = 



OZ'NM 

OA '■ 



or z = -Xj where x^ 
a 



t 



P is on curve of intersection of the cyUnders. 

It is seen that the volume found is equal to the volume 
outside the planes of Ex. 9. 

Example 11. A torus is generated by a circle of radius h 
revolving about an axis in its plane, a being the distance 
of the center of the circle from the axis. 




Find the volume by means of sections perpendicular to 
the axis. 

Aydy = I [t (a -\-xy-Tr{a- xY] dy 

hi Jx=-h 

= TT P~^ [(a + V62 - 2/2)2 -{a- V62 - y'')^] dy 
= 7r / 4:aVh^ -y^dy 

|V6^";=:^ + ^6^sin-i^J^^ 

.2} 



= 47ra 
= 47ra 



2Tr^a¥ = 27ra.7r62. 




282 INTEGRAL CALCULUS 

Note. — The last form of the result shows that the volume 
is the product of the area of the cross section and the length 
of the circumference described by the center of the revolving 
circle, radius a being mean of a + 6 and a — b. 

EXERCISE XXXI. 

1. Find the volume of the right conoid whose 
base is a circle of radius a, and whose altitude 
is h. 

(a) With origin at 0, on the circumference; 
y^ = 2ax - x\ 

(6) With origin at C, center; y^ = a^ — x^. 

Ans. ^-. 

2. An isosceles triangle moves perpendicular to the plane of the 
ellipse x^/a^ + y^/V^ = 1, its base is the double ordinate of the ellipse, 
and the vertical angle 2 A is constant. Find the volume generated by 
the triangle. . 4 ab^ cot A 

o 

X? ifi Z^ 

3. Find the volume of the ellipsoid -^+r^ + i = Iby considering 

the volume generated by moving a variable ellipse along the axis of X. 
Area of ellipse = irab. From result get volume of a sphere. 

Ans. jTrabc. 

4. A football is 16 inches long and a plane section containing a seam 
is an ellipse the minor axis of which is 8 inches in length. Find the 
volume (a) if the leather is so stiff that every cross section is a square; 
(6) if the cross section is a circle. Ans. (a) 341^ cu. in. 

(6) —^ cu. m. 

5. To fell a tree 2 a feet in diameter, a cut is made halfway through 
from each side. The lower face of each cut is horizontal; the inclined 
face makes an angle of 45° with the horizontal. Find the volume of 
the wood cut out. Compare Ex. 9 of illustrative examples. 

Ans. J a^ cu. ft. 

6. Find the volume of the eUiptic paraboloid 2x = — + - cut off 

by the plane x = h. Ans. ir "^pq h"^. 

7. Find the volume of Ex. 9 by moving the trapezoidal section along 
the F-axis. Note that the triangular section of the volume outside the 



PRISMOID FORMULA 283 

cutting planes will at the same time generate that volume, the same as 
the volume of 5 above, when h = a. 

8. A cap for a post is a solid of which every horizontal section is a 
square, and the corners of the square lie in the surface of a sphere 12 
inches in diameter with its center in the upper face of the cap. The 
depth of the cap is 4 inches. Find the volume of the cap. Compare 
Ex. 7 of illustrative examples. 

Ans. 490f cu. in. 

9. Find the surface of the cap of 8, above. Compare Ex. 8 of 
illustrative examples. 

Ans. Curved surface = 192 sq. in.; surface of top = 72 sq. in. 

10. Show that the volimae of the frustum of any pyramid or cone is 

equal to ^ (Aq + Ah + VAoAh) where Ao and Ah are the bases, and h 
is its height. 

158. Prismoid Formula.* — If two solids contained 
between the same two parallel planes have all their corre- 
sponding sections parallel to these planes equal, that is, if 
the area A/ of the one is the same as the area A/' of the other, 
then their total volumes are equal, since the two volumes 
are given by the same integral. Let the distance between 
the bounding planes be, in general, s = x, or y, or z. 

If the area A^ is a section of a solid included between two 
parallel planes and is a quadratic function of s, 

As = as^ + 6s + c, (1) 

where s is the distance of the section As from one of the two 
parallel planes, then the volume is given by 

]s=h f*s=h r ^3 o2 -[s=h 

= J ^ (as'' + hs + c)ds = \a^^-\-b^ + cs\^ 

= -3- + "2- + c/i, (2) 

where h is the distance of the terminal plane from the initial 
plane of reference; that is, the height, or length, of the solid, 
as the case may be. 

* This derivation of the formula is substantially that given in Davis's 
Calculus. 



284 INTEGRAL CALCULUS 



The area Ao =As 
the area Ah =As 



= as^ -\- hs -\- c 

s=0 

= as^ + bs + c 

s=h Js=h 



- ah^ -\-hh-\-c) 



and the area i4.^ = A8 =as^ + hs-}-c\ =-r+7r + c, 

where Am is the area of a section midway between the end 
sections, Ao and Ah. 
The average of Ao, A^, and 4 times A^, is 

g (Ao + Aa + 4 A^) = — + — + c; 

and this average section multiplied by h is the total volume : 

Ao+A;,H-4A^^^ , a¥ . hhj^ . , . ._. .^, 
g X/i=-^ + ^ + c/i,asm(2). ... (3) 



Js=0 



This is the Prismoid Formula, so called because it holds 
not only for every solid whose volume is given in elementary 
geometry hut for any prismoid, that is, for a solid with any end 
sections whatever, with sides formed by straight Hues joining 
points of one end section with points of the other end section. 

159. Application of the Prismoid Formula. — The for- 
mula holds even for many solids that are not prismoids, for 
example, spheres and paraboloids. It holds for all solids 
defined by equation (1), Art. 158, and even for all cases 
where A a is any cubic function of s: 

As = as^ + 6s2 ^cs + d. (1) 

When / (x) is a quadratic or a cubic function of x; then, 
in general. 



£;V(.).. = [/(«)+4/(^^)+/(.)]V' 



(2) 



in accordance with the prismoid formula. The practical 
appHcation of the formula is mainly for the close approxi- 
mation it gives to the volume of objects in nature; for any 



APPLICATION OF THE PRISMOID FORMULA 285 



elevation or irregularity of the crust of the earth can be 
approximated to quite closely, either by the frustum of a 
cone, sphere, cylinder, pyramid, paraboloid, wedge, or prism; 
and as the formula holds for these solids as well as for any 
combination of them, it can be applied without determining 
which of the solids actually approximates most nearly to 
the object whose volume is desired. While it is thus used 
to approximate to the volumes of irregular soHds, it is to 
be remembered that it gives the exact volume, when the area 
of a section As is either a quadratic or a cubic function of s, 
including of course a linear function as a special case of the 
quadratic or cubic function. 

Example 1. — In the case of the cone or pyramid, Art. 155, 

it is seen that A^ = A;i 7^ is a quadratic function of x, and 



hence V = ^(o + A, + 4^)/i = ^ A,/i. 

Example 2. — In the case of the sphere. 



Ax = Aq 



T (a^ — x^), 



hence, y = |(0 + + 4Ao)2a = ^ Aoa = f ira^, where Aq 
= Am = Tra^. 

Example 3. — In the case 
of the paraboloid of revolu- 
tion, about the axis OY, of 
the curve y = x"^; 

7=i(0 + A, + 4AJ/i 

Here As = Try is a Hnear 
function of the distance y, 
for by (1), Art. 158, a = 0, 
h = TT, c = 0; hence the 
formula holds. 




286 INTEGRAL CALCULUS 

Example 4. — The prismoid shown in figure is composed 
of a prism, a wedge, and two pyramids. Let Aq be the 
smaller end section, Ah the larger, and Am the mid section. 

V = Aoh = -(Aq-\-Ao + 4: Ao), for prism, 
y = a; I = ^(o + a; + 4 ^^), for wedge, 
(o + A'jI + -—^j, for pyramid. 



V-A"h-^ 



The formula is seen to hold for the three forms of solids 
composing the prismoid. 



As a practical case, let the figure represent a section of a 
railway embankment 100 feet in length. 

A fill of 10 ft. with side slopes 1 J to 1, makes Ao = 250 sq. ft. 
A fill of 20 ft. with side slopes IJ to 1, makes Ah = 800 

sq. ft. 
A fill of 15 ft. with side slopes IJ to 1, makes 4:Am = 1950 
sq. ft. 
Hence, 7= H^ (250 + 800 + 1950) = 50,000 cu. ft. 



APPLICATION OF THE PRISMOID FORMULA 287 

Here V = h/2 (Ao + Ah) = H^ (250 + 800) = 52,500 cu. ft., 
by average end areas. 

V = hAm= 100 X 487.5 = 48,750 cu. ft., by mean area. 
It is seen that the error of the approximation by the average 
end areas is twice that by mean area and of opposite sign. 
Since the errors vary as the square of the difference in dimen- 
sions of the two end areas, when the end areas are very 
different, the true prismoid formula should be used, but 
when the end areas are ahke, or nearly so, the approximate 
formulas may give results as nearly exact as may be desired. 

EXERCISE XXXII. 

1. Get the volume of a frustum of a solid included between the 
planes s = and s = h, when the area As of a parallel cross section is a 
cubic function, as^ + hs^ + cs + d, of the distance s from one of the 
bounding planes; first by direct integration using the frustum formula, 
then by the prismoid formula. Thus prove the statement at the 
beginning of Art. 159. 

2. Show according to (2) Art. 159, as in the case of volumes, that the 
area under any curve y = f (x), where / (x) is any quadratic or cubic 
function of x, between x = a and x = b, is 

—Q-^(ya-\-yb + 4:ym), (1) 

where ya, yb, ym represent the values of y at x = a, x = b, and x = 
I (« + h). 

3. Find, first by direct integration, and then by (1) of Ex. 2, the 
areas under each of the following curves. 

(a) y = x^, between x = and x = 2. 

(6) y = x^ -\-2x -JrS, between x ^ 1 and a; = 5. 

4. Show that, when (1) of Ex. 2 is used to get area under the curve 
y = x^ between x = 1 and x = S, the error is about 4.2 per cent. 

5. Find the volume made by revolving the area between the curve 
y = x^ and the x-axis about the a:-axis, between x = and x = 2. See 
Ex. 3, Art. 159. Find first by (1) of Art. 153; then by the prismoid 
formula show that the result by that formula is in error about 4.2 
per cent. 

Note. — The prismoid formula is not applicable for exact results, 
when As is given by a higher function than a cubic; in that case, it and 
the general formula (2), Art. 159, for / (x), give approximations. 



288 



INTEGRAL CALCULUS 



160. Surfaces and Solids of Revolution. — To get an 

expression for the area of a surface made by the revolution 

of a curve y = f (x) 
about the axis OX, let 
-Po {xo, 2/0) be a fixed 
point and P {x, y) a va- 
riable point on the curve 
OPoP. Let PoP = s, 
and PP' = As, and let 
PD and P'R be drawn 
each parallel to OX and 
equal in length to As. 

Let S denote the surface generated by the revolution of PqP 

about the x-axis; then A>S equals the surface generated by 

PP'. It is evident that 

surface PD < A>S < surface P'/2; 

that is, 2'Ky^s<^S<2^^{y-\- \y) As; 

A.S 




dividing by As, 



27^y<^<2'I^{y-\-^y)', 



hence, limit | -^ = -7- = 2 tt^/, since A?/ = 0, as As = 0; 



As=0 



LAs 



dS = 2Try ds or S 



= 2ir r 

Jo 



yds. (See (3) Art. 157.) (1) 



Here dS = 2 7ry ds may be represented by the lateral surface 
of a cylinder MPT', the circumference of whose base is 2 iry 
and whose length is PT', drawn parallel to OX and equal to 
PT, which represents ds along the tangent at P. This is so, 
for this surface is what the change of >S would be, if at P the 
change became uniform, ds being* the uniform change of s 
as X increases uniformly from that point. The surface S 
may be considered as generated by the circumference of a 
circle of varying radius y and hence the point P moving on 
the curve according to the law expressed by its equation 
y=f{x). Since 



SURFACES AND SOLIDS OF REVOLUTION 289 

ds = idx^ + dy¥ = [l+{fjjdx or [{^J + l] dy, 
(1) becomes 

Similarly, when the y-axis is the axis of revolution, 

These formulas may be derived as the limits of sums; thus, 
S = ^ (surfaces As) = lim 2) (surfaces chord As), 



As=0 

/ arc As 
V chord As 

Aa; 



= lim y 2 Try As = lim T 2 tt?/ 1 + 

Ax=0 ^0 Ax=0 ^Xo L 



m' 



The other forms may be derived in the same way, which is 
an abbreviation of a rigorous derivation. 

In any particular example to which these formulas are 
applicable, use that form which involves the simpler inte- 
gration. 

For volumes of sohds of revolution; 

y = TT r%2 ^x = TT r (/ {x)Y dx, (See (4), Art. 157) (6) 
when the revolution is about the a^-axis; and 

V = T f\'dy = ir f\f(y)ydy, (7) 

when the 2/-axis is the axis of revolution. 



290 INTEGRAL CALCULUS 

A derivation as the limit of a sum has been given in Art. 
153. In the figure of this Art. 160, if V is the volume made 
by the revolution of the area MqPqPM about OX, then dV 
is the volume of the cyhnder MPN, whose base is iry^ and 
whose length is PN = dx. This is so, for this volume is 
what the change of the volume V would be, if at P the change 
became uniform, as x increased uniformly by l^x = dx from 
that point. As in the case of the surface S the volume V 
may be considered as being generated by a circle of varying 
radius y, the center of the moving circle always on the a;-axis 
and the point P moving on the curve according to its equation. 

By the method of limits, it is evident that, if P'R = As, 

volume MM'P' > AF > M'MP; 

that is, Triy -\- AyY Ax > AV > wy'^ ^x; 

AV 

dividmg by Ax, tt (y -\- Ayf > ^ > ^2/^> 

hence, lim -r— = ^r- = 'tt/^, since Ay = 0, as Ax = 0; 
Ax=oL^^J dx 

.*. dV = iry'^dx or V = t f y^ dx. (6) 

If the revolution is made about a line y = h, then 

V = 7r f\y-hydx, (8) 

and when the revolution is about a line x = a, then 

y = 7r f\x-aydy. (9) 

Note. — It may be noted that the cone and the sphere of 
Art. 155 and the paraboloids of Art. 156 and Art. 159 are all 
solids of revolution, and hence the formulas of this Art. 160 
are apphcable to the determination of their surfaces and 
volumes. 



SURFACES AND SOLIDS OF REVOLUTION 



291 



Example 1. — Find the volume generated by the revolu- 
tion of the area of the equilateral hyperbola xy = 1 about 
OX. 

\_Xo Xjx=x, 

hence, the entire volume has no limit. 

r 1 l>°=i 
y = 7r = IT cubic units; 

\_Xo Xja;=oO 




hence the limit of the volume, from the section at Xq = OMq 
= 1, extending indefinitely to the right, is the same as the 
volume of the cylinder generated by the revolution of NPq, 
the abscissa of Po, about OX. Thus, while the area under 
the curve y = 1/x, from the ordinate MqPq at x = 1, in- 
definitely to the right, is unlimited (as shown in Ex. 11, Art. 
135), the volume made by its revolution about OX has a 
definite limit. According to Art. 156, if the curve y''=Tr/x 



292 INTEGRAL CALCULUS 

is drawn, any one of its ordinates in linear units will represent 
the volume of the solid extending indefinitely to the right of 
that ordinate; thus, in the figure the ordinate MqPq" = tt 
represents the volume to the right of PqMqPo', and the 
ordinate MiPi" = i tt, the volume to the right of PiMiPi. 
In general, the ordinate MP'' at x = OM represents the 
volume of the soHd to the right of the section at any distance 
X from the origin, and it represents also the area under the 
curve y = ir/x^ to the right of the ordinate to that curve. 

Example 2. — Find the volume to the left of the iz-axis of 
the solid generated by the revolution of the exponential curve 
^ = e^ about the ic-axis. 

y^dx = IT f e^x ^^ == g2a; = cubic units. 
(See Ex. 2, Art. 130, for figure.) 

EXERCISE XXXm. 

In these examples, a segment of a solid of revolution means the 
portion included between two planes, perpendicular to its axis, the solid 
or its segment being, in general, a frustum; and a zone means the convex 
surface of a segment. 

1. Find the area of a zone of the paraboloid of revolution about the 

a:-axis . y^ = 2 px, the plane curve . Ans.' -^[(p'^-\-y^)^ — (p^+ y^?) ^] . . 

6 p 

See Ex. 2, Art. 157, where p = i, yo = 0. 

2. Find the area of a zone of the ellipsoid of revolution about the 

rr-axis; that is, a zone of the prolate spheroid. Get entire surface. 

52 
2/2 = — (a2 — a;2) = (1 — e^) (a2 — x^), where e is the eccentricity. 



y ds = - V a^ — e'^x^ dx. 



.-. S = 2t- C Va^ -eH^dx 
a Jx 

= ir-\x Va2 - 6^x2 + - sin-i-T . 
a L e a Jxo 

The entire surface = 27r?) [6 + ia/e) sin~i e]. 



SURFACES AND SOLIDS OF REVOLUTION 293 

The surface of a sphere = limit 2 7r6 [6+ {a/e) sin"^ e] = 2 ira [a+a] = 4 Tra^, 

since for circle e = 0, limit ^^ — - = lim - — = l ; o = 6. 
e=o L e J e=o Lsm^J 

3. Find the area of the smiace generated by the revolution of the 
cycloid about its base. 

Taking the parametric equations of the cycloid, 

X = a(d — sine), y = a (1 — cos6); 
dx = a{l — cos 6) dd, dy = a sin 6 dd; 
ds = Vdx^ + dy^ = a V2 (1 - cos d) dd. 



S = 2Tjyds = 2Ta''J^''V2(l-GOseydd = 16wa^Jsm'(^d(^-- 

4. Find the surface generated by revolving the catenary about the 
y-SLxis, from x = to x = a. Also about the a:-axis. 

Here y = ^\e^ -\- e~^) , ds = ^ \e« + e~ y dx. 
S = 2xCxds==Trf a: \e« + e ^)dx 

r /^ _A .a/f _E\ 1 

= Tr\x'a\e^ — e ^/— a J \e° — e °/ dx , by parts, 
= TT ax f e« - e~ ^) - o? \e« + e~ "j = 27ra2 (1 - g-i). 

About X-axis: ;S = 27r j ?/ds = 2 7r J - \e^ + e °y dx 

= TT I (e^ - e~ """j + ax = ^' (e^ - e-2 + 4). 



5. Find the entire surface generated by revolving the h3rpocycloid 
about the x-axis. x^ + t/^ = c^ is the equation of the curve. 

J« 12 



6. Find the area of a zone of the surface generated by the tractrix 
revolving about the x-axis. (See Art. 150.) 

^ = 27r J 2/ rfs = 27r J% (- ^)= 27ra [-?/]" = 27ra (2/0 - 2/). 



294 INTEGRAL CALCULUS 

7. A quadrant of a circle is revolved about a tangent at one extrem- 
ity. Find the area of the curved surface generated. 

S = 2Tf{a-x)ds = 2x j"" {a-x)(l + -y dx, 

when tangent to x^ + 2/^ = cfi is perpendicular to x-axis, 
r r"' o?dx _ r"' xdx ~| 
= 2^^ Uo Vo? - x^ -^0 Va2 - x^i 

= 2 TT fa' sin-i - + a Vo^ - x^l "" = tto? (tt - 2). 
V. ci Jo 

8. Find the volume of a segment of the prolate spheroid, and the 
entire volume. Find the latter to be two-thirds the volume of the cir- 
cumscribed cylinder of revolution. 

Ans. ~\o?{x — Xq)—^ (x3 — a^o^) • 

9. Find the volume of the oblate spheroid, that is, the ellipsoid of 
revolution about the minor axis which is on the z/-axis. 

Find the volume to be two-thirds of that of the circumscribed cyl- 
inder of revolution. 

10. Find the volume of the paraboloid made by a;^ = 2 j>y about the 
?/-axis. (Compare Ex. 3, Art. 159.) 

Find the volume to be one-half that of the circumscribed cylinder of 
revolution. 

11. Find that the volume of the solid generated by revolving an 
arch of the cycloid about its base is five-eighths of the circumscribed 
cylinder. 

Here V = 2^^ C"' —£^=^ or V = -wa^ C"" {I - co^eydB. 

•^0 ^2 ay — y"^ -'o 

12. Find the volume generated by the catenary revolving about 
the a;-axis, from x =^ a \>o x = —a. Also find the volume by the area 
with the same arc revolving about the ^-axis. 

Here F = tt f ~\f ^ e <') dx = ^-\%e'' ^2x -%e M 

J —a 4 4 \_Z £t J— o 

= "^ (e^ + 4 - e-2) = 8.83 a\ 

/ X x\ 

And 7 = TT f x^dy^'^-^ ^2 Ve« - e~ «/ dx. 

J Q 2i «/o 



SURFACES AND SOLIDS OF REVOLUTION 295 

Integrating by parts gives ^-^ 

y =^ ax^^e" +e V - 2a^x\e^ - e~ ~^ ) -\- 2 a^ \e^ +e~A 

= ^' (e + 5 e-i - 4) = 0.878 aK 

13. Find the volume of the solid generated by the revolution of the 
tractrix about the a:-axis. 

Xoo fO , -jrnS 

y'dx= -w i Va^-y2ydy='^- 
'fa 6 

14. Find the volume generated by the revolution of the hypocycloid 
about the x-axis. 

//\a on 

if dx = -K I (a^ — x^)"^ dx = —prz-KO?. 
J— a 105 

15. Find the volume generated by the revolution about the ?/-axis 
of the equilateral hyperbola xy = 1, from x = to a: = 1. 

F=7r| x^dy ^ -K \ -^=— - =7r cubic units. 

J:c=o "^ Jy=\ y^ yJi 

(Compare Ex. 1, Art. 160.) 

16. Find the volume of the segment of the solid generated by the 
revolution of the equilateral hyperbola x'^ — y^ = o? about the x-axis, 
the altitude of the segment being a, measured from the vertex. 

Ans. f Tra^ 

17. Find the volume generated by revolving about either axis the 
part of the parabola x^ + y^ = a^ intercepted by the axes. 

Ans. yV Trav- 
is. Find the volume of the solid generated by the quadrant of a 
circle revolved about a tangent at one extremity. 

V = irj\a - xydy =-K ^\a- V^f^^f dy = wa' (5 _ "0. 

19. Find the volume generated by the revolution of the cissoid 
2/2 = - — — — about the a:-axis, from the origin to a; = a. 

Ans. |7ra3(31og2 - 2). 

20. Find the volume generated by the revolution of the cissoid about 
its asymptote x = 2 a. Ans. 2 ir^a^. 



CHAPTER V. 

SUCCESSIVE INTEGRATION. MULTIPLE 
INTEGRALS. SURFACES AND VOLUMES. 

161. Successive Integration. — As the inverse of succes- 
sive differentiation there is successive integration. If a 
start is made with a function y = f (x), considered as an nth 
derived function, a single integration gives another function, 
the integral; the integration of this function gives a second 
integral, and so on. The result of n integrations is the nth 
integral of the given function. 

In Art. 140 on Integral Curves successive integration was 
indicated, and in Art. 141 the process was employed in 
application to beams. For successive integration with 
respect to a single independent variable, in general; let 

hix)=Jj{x)dx, (1) 

f2(x)=JfUx)dx,- (2) 

fz(x)=Jf2{x)dx. (3) 

Since /2W = / lfiix)]dx, 

it follows from (1) that 

f2{x)=^J^ff{x)dxyx; (4) 

and since fsix) = / [f2{x)]dx, 

296 



SUCCESSIVE INTEGRATION 297 

it follows from (4) that 

f^(^)=f\f [ff (^) ^^] ^^ I ^^- (^) 

The integral in (4) is called a double integral and is written 



JJf{x)dx\ 



Similarly, the integral in (5) is called a triple integral and is 
written 



/// 



fix) dxK 

If an integral is evaluated by two or more successive in- 
tegrations, it is called a multiple integral. 

For example, to evaluate the multiple integral I I I e'^dx^; 

e^ + C\ is the first integral, 

6=^ + CiX + C2 is the second integral, 

CxX^ 

e"" H — - — h CiX + C3 is the third integral. 
Hence T f fe- dx^ = e- + ^ + C^x + C3. 



If Hmits are given for each successive integration, the 

integral is definite; if Hmits are not given, it is indefinite. 

d'^s 
Example 1. — Given the acceleration -tj^ = —g to find s. 

This is Ex. 5 of Art. 115, and may be written thus: 
s=^JJ-gdt\ 

s = I ( — gt + vo) dt, where Vq is the constant of integration, 

s = —\gt^-\-VQt-\- So, where So is the constant of integration. 

Example 2. — Determine the curve for every point of 
which the rate of change of the slope is 2. 



298 INTEGRAL CALCULUS 

TT d^y _ d /dy\ _ dm _ 

dx^ dx \dx) ~ dx ~ ' 

.*. y=fj2dx^ 

y = I (2 X + Ci) dx, where 2x + Ciis the first integral, 
y = x^ -\- Cix + C2, the second integral. 

This is the equation of any parabola that has its axis parallel 
to the y-axis and drawn upwards, and its latus rectum equal 
to 1. All such parabolas may be gotten by giving all possible 
values to Ci and C2, the arbitrary constants of integration. 

Example 3. — Determine the locus of the equation ;j-^ = 0. 

y=JJodx% 

y = I mdx, where m is the constant of integration, 

y = mx + b, where h is the constant of integration. 

The locus is the system of straight lines, the arbitrary con- 
stants m and h representing the slope and ^/-iJ^tercept, 
respectively. 
Example 4. — In the theory of flexure of beams 



dx^ E/r ^^^ 2 



where E, I, M, R, and w are constants. Get an expression for 
y and dexermine the constants of integration from the con- 
ditions, y = when x = 0, and y = when x = L 



SUCCESSIVE INTEGRATION 299 

" ^' Ell 2 6 "^24 J' 

^ _]_ VM^ Rx^ wx^ Ml RP win 
'^ ~ Ell 2 ^ Q 24 2 6 "^24]' 

/•2 f*3 PA 

Example 5. — Evaluate III x^dx^. 

t/O Ji J2 

Letting I denote the integral and making the integrations 
in order from right to left; 



I 



60 rM'da; = 120 Pdx = 240. 



Example 6. — A point has an acceleration expressed by 
the equation at = — rco^ sin o)t, where r and co are constants. 
Get expressions for the velocity and the distance or space 
passed over. 

d^s c r 

Here a^ = -p = — rco^ sin oit and s = j / — rco^ sin ot dt^, 
.'. ^ ~ ;77 ^ / ;j72 ^^ ^ ~ ^^^ / ^^^ ^^ ^^ ^ ^^ ^^^ ^^ + ^i> 

s = I -r.dt = ro) j cos co^ di + I Ci rf^, 
s = r sin co^ + Ci^ + C2, 
which is the law of simple harmonic motion. (See Art. 73.) 

EXERCISE XXXIV. 

1. Evaluate fjf {x' - 1) dx\ Ans. ^ - f + ^' + ^'2^ + C,. 

2. Evaluate J J JJ*^da;4. 

Ans. -i log X + i x3 + I C2X2 + C3X_+ 04. 



300 INTEGRAL CALCULUS 



3. Evaluate lilt sinaxdx^ 



//// 



Ans. l/a^ sin ax + ^ Cix^ + I CiX"^ + Czx + C*. 

4 /»3 /.2 



4. Evaluate f f f a^^t^^^, ^^s^ 16 

»'2 Jl ^0 

5. Evaluate f f f x^dx^. Ans. 80 

•/2 ^0 *^1 



'0 

6. Find the curve at each of whose points the rate of change of the 
slope is four times the abscissa, and which passes through the origin 
and the point (2, 4). Ans. Sy = 2x{x^ - I). 

7. Evaluate r^ r fsin^d^'. Ans.wifi — a). 

Jo J a Ja 

d^s 

8. The differential equation of falling bodies is -t-^ = —g; show 

that s = -~ -\-Cit + C2', and find Ci and d, if s = and y = 100, 

A 

when t = 0. 

9. A point has an acceleration expressed by the equation at = 
—ru? cos (Jit, where r and w are constants. Find expressions for the 
velocity and the distance passed over. Find C\ and C2, if s = r and 
e; = 0, when i = 0. 

Ans. V = — rco sin coif + C'l; s = r cos wt + Cii + C2 

162. Successive Integration with Respect to Two or 
More Independent Variables. — In the preceding Article 
successive integration was of functions with respect to a 
single independent variable. Successive integration of 
functions of several independent variables are now to be 
considered. Suppose there is given a function / (x, y, z) of 
three independent variables. 

Let /i (x, y,z)=jf {x, y, z) dz, (1) 

/2 (x, y,z)= j /i {x, y, z) dy, (2) 

/3 (x, y,z)=J /2 {x, y, z) dx, (3) 

where in (1) the integration is with respect to z, that is, as if 
x and y were constants. Likewise in (2) it is with respect 



THE CONSTANT OF INTEGRATION 301 

to y, as if x and z were constants, and in (3) with respect to 
X, as if y and z were constants. 

Equation (2), by substitution from (1), becomes 

/2 (x, y,z) = J \jf {x, y, z) dz\ dy; (4) 

and equation (3), by substitution from (4), becomes 

/s (x, y^ ^^ == J ] J \J (^' y^ ^) ^^J ^y i ^^- (^) 

The integral in (5) is called a triple integral and is written 
jjjf{x,y,z)dxdydz, (6) 

where the order of the integrations is from right to left; that 
is, the differential coefficient / {x, y, z) is to be integrated 
with respect to z, that result to be integrated with respect to 
2/, and finally the last result is to be integrated with respect 
to x. 

Similarly, the double integral in (4) is written : 



// 



fix, y, z)dydz. 

As to the integration signs, the first on the right is to be taken 
with the first differential on the right, which is dz in (6), the 
second sign from the right with the second differential from 
the right, and so on. 

If when limits of integration are given, they are constant 
hmits, the order of the integrations may be reversed without 
affecting the result, but when the definite integral has variable 
limits the order of the integrations can be changed only by 
new limits adapted to the new order. In practical problems 
the limits for one variable are often functions of one or more 
of the other variables. 

163. The Constant of Integration. — The evaluation of 
an indefinite multiple integral differs from that of an indefi- 
nite single integral in the form of the constant of integration. 



302 . INTEGRAL CALCULUS 

Thus / I 4:xy dxdy being given, to find a function u oi x 
and y such that 

is the problem. It is evident the operations represented by 
, , dx dy must be reversed in order to get u. 

That is, u = j I -7—7- dxdy = j I 4:xydx dy, (1) 

which indicates two successive integrations, the first with 
respect to y, x and dx regarded as constants, and the second 
with respect to x, y being regarded as constant. Hence the 
first integration gives 

-7- = 2 xy^ + constant of integration. 

Since x was regarded as constant during the integration, 
the constant of integration may depend upon x, that is, it 
may be some function (j){x), or it may be simply C. This is 
so, since differentiating either 2 xy^ + C, or 2 xy^ + </> (a:), 
with respect to y gives the same result, 4 xy. Hence, 

where (x) is an arbitrary function of x and may be a con- 
stant C. 

Integrating this result, with y constant, gives 

u = xY + J<i> ix) dx + F (y), (2) 

where, since y was regarded as constant during the integra- 
tion, the integration constant is an arbitrary function of y 
and may be C with a constant value, possibly zero. 



THE CONSTANT OF INTEGRATION 303 



u = x^ -\- xV; ,^ 7 dxdy = 4: xy dx dy\ 



By referring to Art. 109, (2), it will be seen that if 

dy 
that is, (/) (oj) = 3 ^2 and F (y) = 0, for that function u of 

fe y)' 
The indefiniteness of the result in (2) is manifest, for 



u = I j 4:xydxdy = x'^y'^ 



if both constants of integration are zero, that is, </> (x) =0 
and F (y) = 0. The indefiniteness is removed when limits 
for the variables are given, the integral being then a definite 
integral. 
Example. — 

I I xyz dxdydz= I j xydxdy\-\ 







-£ 


^xdx r- , „ 

2 Jo^(^" 


- y') dy 






-£ 


'x^ . 21 „ 






EXERCISE XXXV. 




Evaluate the following integrals : 






1. Jfx^ydxdy. 






Ans. \::?y^^F{x) -\-fi{y), 


Jo Jo 


'-y dx dy. 






Ans. ^5 



) p2 sin ddpdd = ^ 

Jo o 



p 
Jo '^'^ 

2 

• 22/ 



I xy dy dx =* ^^ 6*. 

Jy—b 



304 INTEGRAL CALCULUS 

TT 

J'»^ /*2acos9 TT 

•/2 6 cose -6 

Xh fy 5^ — a? 

J p2 sin edpdd = — - — (cos - cos y), 

ph plot 

8. 1 I Vst-t^dtds = 6¥. 
Jo Jt 

n2 /•S 
j xy'^ dx dy dz = 17i 

10. f r f xy"^ dz dy dx = 24|. 

J2 J\ •'2 

11. ^ { ^ xy^ dz dy dx = 17|. 

%/2 •'1 •'2 

12. r f r a^V^Jf^ajfii/fis = ia362(63 _a3). 

( r e'^+y+'dxdydz=^—^ -~-^e\ 

. Jo Jo o 4 

14. f ^ C^ V2^ dxdy = l V2~g {h^ - hx^) b. 
Jo J hi 

Jo J a(l— COS0) o 



16. r C {lo ■\-2v)dvdw= -W- a'. 

17. r^" r f^'^aj^ ^2 dx dy dz = 32 a', j 

./O •'0 »/2y 

164. Plane Areas by Double Integration — Rectangular 
coordinates. — It has been shown in Art, 135, that the area 
between two curves y = f(x) and y = F (x) is given by 

A= r\f{x)-F{x))dx, (1) 

where the points of intersection are (xq, 2/0) and {xi, yi) . The 
area is thus given not by a single integral but by the differ- 
between two integrals, / f{x)dx — \ F (x) dx. The 

result is gotten also by double integration, finding the hmit 
of two sums. Let the element of area be A?/ Aic, (ic, y) being 
any point P of the area. If the elements are summed 



ence 



PLANE AREAS BY DOUBLE INTEGRATION 305 



up with respect to y, with the Hmits MD and MN, or F{x) 
and f {x), X being constant, the area of the strip DN' is 
gotten. If the strips are summed up with the hmits a and 
h for X, then 

x=a\_F{x) J "-Fix) 

is the expression for the sums. Taking the hmits of the 

Y 




sums, first as Ai/ = 6 and then as Lx = 0, the area ADEN 
is given by the double integral 



Pb nf{x) 
A = / I dxdy, 

Ja JF{x) 



(2) 



which integrated first with respect to y gives 

A=£{f{x)-F{x))dx. (1) 

If the elements are summed up in reverse order, first with 
respect to x with the limits H'H and H'S, or f~^{y) and 
^~^{y)j y being constant, and then with respect to y with 
hmits c and d, there results 

*F-Hy) 

dy dx, (3) 



Jc Jf-i 



(y) 



306 INTEGRAL CALCULUS 

where f~^ (y) and F~'^ (y) are the inverse functions of / (x) and 
F {x), respectively. 

Integrating (3) the area ADEN is gotten, as given by (2). 
Hence, in general, 

A= f fdx dy (4) 

is the formula for area by double integration, the limits being 
taken so as to include the required area. The order of 
integration is indifferent provided the limits be adapted to 
the order taken. 

Corollary. — dxy^A = dx dy and dyx^A = dy dx. 

Example 1. — Find the area bounded by the parabolas 
y^ = 2px and x^ = 2 py. 

The parabolas intersect at the points (0, 0) and (2 p, 2 p). 

I dxdy = ^ p^, by formula (2). 

J x2 



Jo J «2 



2p2/ 

dydx = ^ p"^, by formula (3). 



2p 

Example 2. — Find the area bounded by the circle x^ + i/^ 
= 12, the parabola y^ = 4 a;, and the 
parabola x^ = ^y. 

For the part OPP2 the limits for x are 
and 2, while for the part PP1P2, they 
are 2 and Vs, the point P being (2, Vs) 
-X and the point Pi (Vs, 2). For both parts 
the lower limit for y is the ordinate of 
a:^ = 4 2/; for OPP2 the upper limit for y is the ordinate of 
y^ = 4:X, and for PP1P2, that of x^ + i/^ = 12. 

A = OPP1P2 = OPP2 + PP1P2 = r r ' dx dy 

Jo J x'i 




/ dxdy 

2 Jx^ 



PLANE AREAS BY DOUBLE INTEGRATION 307 



Example 3. — Find the area between the parabola 
and the circle y^ = 2ax — x^. 



ax 



nv2ai-xa 
dxdy 
^. 

= 21 iV2ax - x2 - Vo^) dx 
Jo 



7ra' 
"2" 



4a^ 
3 



Note. — It may be seen that in finding some areas there is 
no advantage in using double integration, as after the first 
integration with the limits substituted, the remaining in- 
tegral is what might have been formed at first. There are, 
however, cases where double integration furnishes the only- 
method of solution; hence the need for some practice in its 
application. 



EXERCISE XXXVI. 

1. Find the area between the circle x^ -\- y"^ = a^ and the line y 
a — X. 



ra /»va2— x2 
A = I I dxdy = 



-2 



2. Find by double integration the area between the parabolas 
1/2 = 8 X and x^ = 8y. 

. Ans. 21i 

3. Find the area bounded by the circle x^ -\- y^ = 25, the parabola 
y^ = -^/ x, and the parabola y = ts ^^• 

Ans. 7.55. 

4. Find by double integration the area of the 

x^ ifi 
ellipse - + p = 1. 

5. Find the area of any right triangle, using 
double integration. 



A- f f dxdy = £ {-Ix + a'j 




dx = — - -7Z +ax \ =7:00. 
Jo I 



h 2 



308 



INTEGRAL CALCULUS 



165. Plane Areas by Double Integration — Polar coor- 
dinates. — As has been shown in Art. 135(b), the area in 

polar coordinates of P1OP2, 
generated by the radius 
vector p as ^ increases from 
$1 to 02 is given by 







(1) 



To find the area between 
two polar curves by double 
integration, let the elenient 
of area be PDD'P', bound- 
ed by the two radii 0P\ 
OD', and the two circular arcs, concentric at 0. 

Let the coordinates of P be (p, 6) ; then from geometry, 

sector POD = i p^ A^, 
sector P'OD' = i (p + ^pY A(9. 

Hence, A^ = PDD'P' = i (p + Ap)^ ^6]- J p^ A(9 
= (p + JAp)A^Ap. 

Keeping A0 constant and summing the elements of area 
with respect to p gives an area AA'B'B, expressed by 

OA r^oA 

AS • lim T (p + i Ap) Ap = A6> / pdp. 

Ap=0 OA' JOA' 

Making the summation now with respect to 6, the sum of 
the radial slices is gotten, and the limit of this sum is 

^^2 fOA roi roA 

A = lim y A^ • / pdp= I I pdpdd. 

Replacing OA' and OA by F{e) and / {B) respectively, the 
formula is 

' I pdedp. (2) 

When F (d) = 0, the area P1OP2 between the curve p = f (d) 



PLANE AREAS BY DOUBLE INTEGRATION 309 



and the radii is (1), A 



p2 do, where (2) has been in- 



tegrated as to p. * 

If the summing of the elements of area be made first with 
respect to 6, keeping Ap constant, RSDP, a segment of a 
circular ring, is gotten. A second summation with respect 
to p gives the sum of such ring segments, the limit of which 
sum is the area A. The resulting formula is 






'(p) 



(p) 



p dp do, 



(3) 



where F-^{p) and/^^p) are the inverse functions of F{d) and 
/ (d), respectively. 

Corollary. — dop^A = pdddp and dpe'^A = pdpdB are rec- 
tangles with sides p dS and dp. 

Example 1. — A simple case of the application of the 
formulas is in finding the area of the circle p = a. 




(2) 



(3) 



I pdddp = i / p^ddl 

t/0 Jo J 

-j27r 

= ia^d\ = Tra 

n27r ra 127r 

pdpde= I pdpdl 
Jo Jo 



310 



INTEGRAL CALCULUS 



In (2) the sectors are summed, while in (3) the rings are 
summed. In this case of the circle it is to be noted that no 

limit need be invoked, 
since the integral is the 
sum in each case, the in- 
crements being the differ- 
entials, the variables all 
increasing uniformly. 

Example 2. — Find the 
area between the two tan-* 
gent circles p = 2a cos 6 
and p = 2 6 cos ^, where 
a>h. 




—X 



/ 

t/26co 



I 







pdddp = 4:{a^- 

= 7r(a2-62). 

Example 3. — Find the areas between the cardioid p = 
2 a (1 — cos 6) and the circle p = 2 a. 




n2a 
pdddp 
_a(l-cos0) 

IT 

= 4a2 j (2cose-cos,^d)dd = Sa^-Tra\ 



AREA OF ANY SURFACE BY DOUBLE INTEGRATION 311 

J^»7r /'2a(l-cos0) 
/ pdddfy 

TT t/2a 
2 

= 4a2 r"[(l -cos(9)2- i]^0 

-2 cos ^ + cos2 d)dd = 80^+ Tra\ 



'^'fy 



EXERCISE XXXVn. 



1. Find by double integration the entire area of the cardioid p = 
2a(l— COS0). Ans. 6 Tra^. 

2. Find the area (1) between the first and the second spire of the 
spiral of Archimedes p = ad; (2) between any two consecutive spires; 
(3) the area described by the radius vector in one revolution from 6 = 0, 
and the area added by the nth. revolution. 

Ans. (1) -2/7r3a2; (2) (n^ + 2 n + f ) Tr^a^ ; (3) fTr^aM (n^ - I)7r3a2. 

3. Find by double integration the area of one loop of the lemniscate 
p2 = a^ cos 2d. Ans. | a^. 

4. Find by double integration the area between the circle p = cos 
and one loop of the lemniscate p^ = cos 2 6. Get the area between the 
circle and the line 6 ■= 7r/4 and then between that line, the lemniscate, 
and the circle. . tt — 2 w — 2 , w — 2 

166. Area of any Surface by Double Integration. — Let 

the surface be given by an equation between the rectangular 
coordinates, x, y, z. Let the equation of the given surface 
be 

Passing two series of planes parallel, respectively, to XZ 
and YZ, will divide the given surface into elements. These 
planes will at the same time divide the plane XY into ele- 
mentary rectangles, one of which is P'P^, the projection 
upon the plane XF of the corresponding element of the 
surface PP2. 

Let X, y, z be the coordinates of P and x + ^x, y + A?/, 
z + Az, those of P2, x and y being independent; then P'M' = 
Ax and P'N' = \y. The planes which cut the element PP2 



312 



INTEGRAL CALCULUS 



from the surface will cut a parallelogram from the tangent 
plane at P, the projection of which on the plane XF is P'P2 
= Ax Ay, the same as the projection of the element PP2. 
The projection is the product of the area of the parallelogram 




and the cosine of the angle made by the tangent plane with 
the plane XY; hence, denoting the angle by 7 and the 
parallelogram cut from the tangent plane by PT, 
area PT = area P'P2 • sec 7 
= Ax Ay secy. 
As Ax and Ay approach zero, the point P2 approaches the 
point P, and the areas PT and PP2 approach equality; that 
is, the element of surface approaches coincidence with the 
parallelogram, a portion of the tangent plane at P; hence, 

areaPP2 = AxJ^S = Ax A?/ sec 7, approximately; 
that is, 



area 



PP. = A./S^A.A,sec.; li^„[^] = see.; ("^^ 



Ay=0 

dxy^S = sec 7 • dx dy. 



) 

(1) 



AREA OF ANY SURFACE BY DOUBLE INTEGRATION 313 



fi 4- /^Y-i- /^Y?. ( ^^^ figure \ 
\S^\dx) ~^\dy) J ' VofArt. lOl.y 



FromArt.l03(8),secT 
hence from (1), 



(2) 



the Hmits being so taken as to include the desired surface. 

"Let S denote that part of the surface z = fix, y), z being 

a one-valued function, which is included by the cyhndrical 

surfaces y = (l^o (x) , y = (j) (x) , and the planes x = a, y = h; 



J a Xo(x) L W/ \dy) 



dxdy. 



(20 



In finding the area of the given surface a more convenient 
form of the equation of the surface may be either x — 
f (y, z), or y = f {z, x). The formula for the area will be 
then either 



//[-(i)'+(s)": 



or 



//[ 



'+(2)'+(2)7 



dydz, 
dxdz, 



(3) 
(4) 



with the proper limits of integration. 

In applying the formulas, the values of the partial deriva- 
tives are gotten from the equation of the surface the area of 
which is sought; hence, when there are two surfaces each of 
which intercepts a portion of the other, the partial derivatives 
in each case are taken from the equation of that surface 
whose partial area is being sought. This will be illustrated 
in the following examples. 

Example 1. — To find the surface of the sphere whose 
equation is 

x^ + y^ -\- z^ = a^. 



314 INTEGRAL CALCULUS 

Let - ABC of the figure (Art. 166) be one-eighth of 
the sphere. 

dz^ _ _x ^ _ y ' 
dx z dy z^ 

,2 



\dx/ \dyj z^ z^ z^ o} — x 

V a 



dxdy 



^ — x^— y^ 



by (2) 



dx 



= 8a f\m-^^^=T 
Jo Va2 - xUo 

A C"' J A I Compare Ex. 2, \ 
= 4:7ra I dx = 4 ira^. ^ .^ ^^^' 

Jo V Exercise XXXIII. / 

Here the integration was over the region OAB, the projec- 
tion of the curved surface ABC on XY plane. The first 
integration with respect to y summed all the elements in a 
strip LL'K'K, y varying from zero to NL', that is, between 
limits and v a^ — ^2 ^ ^j^g equation of the intersection of the 
surface with the XY plane being x'^ -\- y'^ = a^. Integrating 
next with respect to x, the surface ABC is gotten by sum- 
ming all the strips from x = io x = a. 

Example 2. — Find the area of the portion of the surface 
of a sphere which is intercepted by a right cylinder, one of 
whose edges passes through the center of the sphere, and 
the radius of whose base is half that of the sphere. 

Note. — This is the celebrated Florentine enigma, pro- 
posed by Vincent Viviani as a challenge to the mathemati- 
cians of his time. (Williamson's Integral Calculus.) 

Taking the origin at the center of the sphere, an element 
of the cylinder for the ;2-axis and a diameter of a right section 
of the cylinder for the x-axis, the equation of the sphere will 
be x^ -\- y'^ -\- z^ = a^, and the equation of the cy Under, 
^2 _|_ ^2 ^ ax. 



AREA OF ANY SURFACE BY DOUBLE INTEGRATION' 315 

The area of A PCD is one-fourth of the area sought, and since 
thiS'^surface is a portion of the surface of the sphere, the par- 
tial derivatives -p , -r must be taken from x^ -\-y^ + z^ = a^, 
ax ay ^ 

giving, as in Ex. 1, using formula (2), 

n _ r r adxdy 

~ J J Va^ -x'-y'- 

to be integrated over the region OP' A. Hence, 

Area = aS = 4 I I ^ = (2 tt — 4) a^. 

Jo Jo Va^ -x^-v^ 




The limits for y are from x^ -\- y"^ = ax, the equation of the 
curve OP' A, the boundary of the projection of the surface 
APCD on the XY plane. 

Example 3. — Find the surface of the cylinder of Ex. 2, 
intercepted by the sphere. 

The area of APCOP' is one-fourth of the area sought, 
and since it is a part of the lateral surface of the cylinder 
x^ -{-y^ = ax, the partial derivatives in formula (2) must be 



316 INTEGRAL CALCULUS 

taken from this equation. But from this equation -—=00, 

dz 

_- = cx) and formula (2) does not apply, which is, more- 

dy 

over, evident since the element of surface is dx dz in the strip 

P'P, and the area of the surface APCOP' cannot be found 

from its projection on the XY plane, for this projection is the 

arc AP'O. The projection is made on the XZ plane and 

formula (4) used. 

The partial derivatives are found to be 

dy ^ a-2x dy^^ 

dx y ^ dz 

Since P is on the sphere, 

pp' = ^2 = ^2 _ (^2 ^ ^2) = ^2 - ax, 
since P is on the cylinder. Hence, 

^-^ = ' = '£sr"v + ^-^11'^'' 

pa rVa^-ax ^^ ^^ CW a^ - ax 

= 2a \ I =2a I 

Jo t/0 V ax — x^ Jo \ax — x^ 



dx 



2a r\/-dx = 4:a\ 



Here the integration is over the region OAP"C, AP"C being 
the projection of APC on XZ plane. The first integration 
sums up the elements of surface in the strip P'P and the 
next integration sums up the strips from x = io x = a. 

By eliminating y from x^ -{- y^ -\- z"^ = a^ and x^ -{- y^ = ax, 
z^ = a^ — ax (as found above), which is the equation of 
AP"C, from which the limits of z are taken. 

EXERCISE XXXVm. 

Find by double integraition the areas of the surfaces given in the 
following examples: 

1. The zone of the sphere, x^ -\- y^ ■\- z"^ = r^, included between the 
planes x = a and x = b. Ans. 2 irr (6 — a). 



VOLUMES BY TRIPLE INTEGRATION 317 

2. The surface of the right cyhnder x^ -{- z^ = a^ intercepted by the 
right cyhnder x^ -\- y^ = o?. Compare Ex. 8, Art. 157. Ans. 8 a^. 

X 11 z 

3. The part of the plane - + f + - = 1, in the first octant, inter- 

a c 

cepted by the coordinate planes. Ans. ^^o?W + a^d^ + ¥c^. 

4. The surface of the cyhnder x"^ -\- y^ = a^, included between the 
plane z = mx and the XY plane. Find by both formula (3) and 
formula (4), and show why formula (2) does not apply. 

Ans. 4 ma^. 

5. The surface of the paraboloid of revolution y^ -\- z^ = 4: ax, 
intercepted by the parabolic cylinder y^ = ax and the plane x = 3 a. 

Jo Jo L 4 ax — 2/2 J 9 

6. The surface of the cylinder of Ex. 5, intercepted by the parabo- 
loidof revolution and the given plane, 

/ \iy_rJ^L±.dxdz = 2VS i {4:ax + a^ydx 

Jo 2y Jo 

= (13V13-1)-^. 
v3 

167. Volumes by Triple Integration — Rectangular Co- 
ordinates. — Let the volume be that of a solid bounded by 
the coordinate planes and any surface given by an equation 
between the coordinates x, y, and z. 

Let P be any point {x, y, z) within the solid ~ ABC, the 
surface being given hy z = f (x, y), where / {x, y) is a con- 
tinuous function. Let K' be the point {x + Ax, y + \y, 
z + A;?), the diagonally opposite corner of the rectangular 
parallelopiped formed by passing planes through P and K', 
the planes being parallel to the coordinate planes. Let more 
planes be passed. Taking first the sum of the elementary 
parallelopipeds whose edges lie along the line NT, the Hmit 
of this sum, as A;2 is made to approach zero, is the volume of 
the prism whose base is ^x Ay and whose altitude is NT, 
X, y, Ax, and At/ remaining constant during the summation. 
Next with x and Ax constant, sum the prisms between the 
planes MHL and SDR. The hmit of this sum as Ay is made 
to approach zero is the volume of the cyhndrical shce 



318 



INTEGRAL CALCULUS 



LR'D'HMS. Finally, when taking the sum of the sHces 
parallel to the YZ plane, as Ax approaches zero, the volume 
of the cylindrical slice approaches that of the actual sHce 
LRDHMS; hence, the limit of the sum of the slices, as Ax 
approaches zero, is the volume of the sohd. 



z 












c 

/ 


> 


L R' 






tX / 


X 


X 






A 




/ 


\ 


\ 


/ 


/ 
/ 


\ / 








\ 




/ / 










\ 




/P' 










\ 


i 


/ y- 










\ 


1 
1 


/-->: 


~K' 










I 




y 










if 


<fi — 


-^ 


y^ 


fy^S 


y 


/ 


y^ 


;/ 











H D'' 



Jf^OA nMH nNT 
I I dxdy dz, 
Jo Jo 



where V is the volume of — ABC. Let V denote the 
volume bounded by the curved surfaces z = fo{x, y), z = 
f{x, y); the cyUndrical surfaces y = <l>o{x), y = 0(x); and 
the planes x = a, x = h; then 

/ dxdydz. (1) 

. J M Jfo {x, y) 

Corollary. — dxyiV = dxdydz, dyzxV = dydzdx, . . . 
If z is expressed in terms of x and y, and /o (x, ^) = 0; 

n4>{x) 
z dx dy. 
. :>{x) 



VOLUMES BY TRIPLE INTEGRATION 319 

Note. — The formula, V = I I I dx dy dz, may be de- 
rived from the figure by the definition of differentials. 

Thus, the variables x, y, z, being independent, dx, dy, dz 
may be taken as finite constants, the parallelopiped PK' being 
dx dy dz. When x and y are regarded constant, PK' is the 
differential of the prism NK. Hence, integrating dx dy dz 
between the Hmits z = and z = NT gives the prism 
NT dx dy, which is the differential of the sohd MSR'L - T. 
Integrating NT dx dy between the Hmits y = and y = MH 
gives the cyHnder MLH — D', or MLH dx, which is the differ- 
ential of the solid OBG — M. Integrating MLH dx between 
the limits x = and x = OA gives the volume OBC — A, or 

V. Hence, V = I I f dx dy dz, the limits being so chosen 

as to include the volume sought. 

Example. — Find the volume of the ellipsoid 

^2 -t- 52 -t- c2 -^• 

The entire volume is eight times that in the first octant, 
where the limits are : 



z = 0, z = cVl- xya^ + yy¥; 
y = 0, y = b V 1 — x'^/a^; 
X = 0, X = a; 

ahc 



/ dxdydz = ^ 

Jo c 

4 

Corollary. — For sphere, a = b = c; .'. V = - 



n a Vl -2/2/52 /»cVl_x2/a2-t/2/62 
/ dy dx dz 

Jo 

/ dz dx dy. 

Jo 



320 INTEGRAL CALCULUS 

EXERCISE XXXIX. 

Find by triple integration the volumes required: 

1. The tetrahedron bounded by the coordinate planes and by the 

plane 

X . y . z ^ . ahc 

ahc 6 

See Ex. 3, Exercise XXXVIII, for the surface of the plane. 

2. The volume bounded by the cylinder x^ -\- y^ ^ o? and the planes 

4 7Y10? 

3 = and z = mx. Ans. — ^ — 

3. A cylindrical vessel with a height of 12 inches and a base diam- 
eter of 8 inches is tipped and the contained liquid is poured out until 
the surface of the remaining liquid coincides with a diameter of the 
base. Find the volume remaining in the vessel, Ans. 128 cu. in. 

Note that the volume is one-half that given by Ex, 2, above. 

4. The volume included between the paraboloid of revolution 
2/2 _|_ 2;2 = 4 ax, the parabolic cylinder y"^ = ax and the plane x = S a. 
See Exs. 5 and 6, Exercise XXXVIII, for the surfaces. 

^sa ^Vax /•(4ax— 2/2)2 . ,-.. 

Ans. 7 = 4) 1 1 dxdydz = iQ7r + 9Vs)a\ 

Jo Jo Jo 

5. The volume included between the paraboloid of revolution 
x^ -\-y^ = az, the cylinder x^ -{- y^ = 2 ax, and the XY plane. 

, x^ + y^ 

r>%a W2ax-x^ f a 

Ans. F = 2 I I I dxdydz = fxa^ 

6. The entire volume bounded by the surface 

^"^■'^^JoJ. Jo dxdydz^-^- 

7. The entire volume bounded by the surface 

I I I I I I A 47ra^ 

x^ -\-y^ +z^ = a^. Ans. -^r^' 

6o 

8. The volume of the part of the cylinder intercepted by the sphere. 
The radius of the sphere is a and it has its center on the surface of a 
right cylinder, the radius of whose base is a/2. See Exs. 2 and 3, 
Art. 166. ' 

I I dxdydz = I {Tr-f)aK 

»/o fc'O 



SOLIDS OF REVOLUTION BY DOUBLE INTEGRATION 321 

168. Solids of Revolution by Double Integration. — In 

the figure of Art. 164, where P{x, y) is any point in the area 
ADEN, X and y being independent, Aa; A?/ is the element 
of area. Conceive the area ADEN to revolve through 6 
radians about OX as an axis; then 

ey . Ax A?/ < A,yW < d {y + Ay) • Ax Ay; 

••• 'y<iS-y<'^y+^y^' 



hence,^ Um . ^^. = -, — ^ = dy ; 

Ax,Ay=o\_AxAy_\ ax ay 

:. dW = dydxdy; 

Jrxi rf{x) 
/ ydxdy. (1) 

Xq JF{x) 

Similarly, about OY, 

Jryi rf-Hy) 
I X dy dx. (2) 

yo Jf-Hv) 

Putting 6 = 2 TT, the formulas give the volumes generated 
by a complete revolution of the area. 

Corollary. — If the axis of revolution cuts the area, (1) 
or (2) will give the difference between the volumes generated 
by the two parts. Hence F = 0, when these two parts 
generate equal volumes. Integrating (1) first with respect 
to y, and (2) first with respect to x, the upper limits being 
the variables y or x and the lower limits zero, gives 

y = ^ r^y^dx (10 

^ Xo 

and y = 7r / x^dy, (20 

the formulas for soHds of revolution, single integration. 

169. Volumes by Triple Integration — Polar Coordinates. 
— Let the point P (p, 6, 0) be any point within a por- 
tion of a sohd bounded by a surface and the rectangular 
planes. As usual, p is the distance OP from the pole at the 
origin, 6 is the angle ZOP which OP makes with the 2;-axis, 



322 



INTEGRAL CALCULUS 



and is the angle XOP' which the projection of OP on the 
XY plane makes with the o^-axis. Let the soUd be divided 
into elementary volumes like PDDiQi by the following 
means. 




(1) Through the 2!-axis pass a series of consecutive planes, 
dividing the solid into wedge-shaped slices such as COAB. 

(2) Round the ;S-axis describe a series of right con^s with 
their vertices at 0, thus dividing each slice into elementary 
pyramids hke - RSTV. 

(3) With as a center describe a series of consecutive 
spheres. Thus the soHd is divided into elementary sohds 
like PDDiQi, whose volume is given approximately by the 
product of three of its edges, PPi, PP2, and PQ. 

Let edge PQ = Ap, angle POP2 = ^6, angle AOB = angle 
PO'Pi = A0; then edge PPi = psind A(/), and edge PP2 = 

p^e. 

Hence, the volume of the elementary solid is given ap- 
proximately by p2 sin 6 AS A0 Ap. It can be expressed 



VOLUMES BY TRIPLE INTEGRATION 323 

exactly but the additional terms vanish when the three in- 
crements are made to approach zero. Therefore, the volume 
of the sohd is given by the limit of 

Z 2 X p'sin^A^A^Ap; 

A0=O A0=O Ap=0 



/. V= I j I p^ sine ddd(l) dp, 



(1) 



each integral to be taken between the limits required to find 
the volume sought. The summation can be made in any 
order so long as the volume is continuous. 

For a solid of revolution with the ^-axis as the axis of 
revolution, the formula (1) for the volume becomes 

V = 2t j I p^ sine dd dp, (2) 

since the hmits for are then evidently and 2 w. The 
limits for p and 6 are then the same as are used in getting 
the area of the plane figure revolved. 

Corollary. — dpe^W = p^ sin 6 dS d(f) dp is an elementary 
rectangular parallelopiped ; and dpeW = 2Trp'^ sin Odd dp is a 
circular ring with rectangular section. 

Example 1. — Find the volume of a sphere of radius a, 
using polar coordinates, pole at end of a diameter. 

By formula (1); or by (2), if the volume is considered as 
generated by revolving the semicircle about the initial fine, 
the line from which 6 is measured, 

2 a cos 

p^ sin 6 do dp 




324 



INTEGRAL CALCULUS 



Example 2. — Find the volume generated by revolving the 
cardioid, p = 2 a (1 — cos 6) about the initial line. 



y = 27r 
167ra' 



J^TT /'2 a (1- 
Jo 



■cos 6) 



p2 sin 6 dd dp 



64 



cos Oy sin Odd = -^-Tra^. 



Example 3. — Find the volume made by revolving the 
lemniscate p^ = a^ cos 2 d about the initial line. 



Jo Jo 



a V cos 2 d 



p^ sin 6 do dp 



4:Td' 

~3~ 

TraM 



J/ ^ ^x 3 • ^ 7o 4 7ra^ 
(cos 2^)2 sm0(i^ = ^r— 
o 

log (V2 + 1) 1\ 



r 



(2cos2(9-l)^sin(9( 



2\/2 



6; 



170. Volumes by Double Integration — Cylindrical Co- 
in finding the volume of some solids the 
integration is performed more 
readily with the use of cylin- 
drical coordinates. 

In this system of coordi- 
nates the position of a point 
is given by the cylindrical co- 
ordinates {r, </), z), where (r, 0) 
are the polar coordinates of 
the projection (x, y, 0), on the 
XY plane, of the point {x, y, z). 

It is evident that the equa- 
tions of transformation from 
rectangular to cylindrical co- 
ordinates are: 




x = r cos 0, 2/ = ^ sin 0, z = z) 



(1) 



VOLUMES BY DOUBLE INTEGRATION 325 

and those from cylindrical to rectangular, 



-iV 



r = Vx^ + 2/2, = cos-i - = sin-i ^ = tan-i ^, z = z. (2) 

To derive a formula for volume the differential element of 
area in the XY plane may be taken as the rectangular base 
of an elementary right prism with altitude z, the base of the 
actual prisms into which the solid may be divided being 
bounded by lines two only of which are right lines, the other 
two being circular arcs, and the altitude of possibly only one 
edge being z, since the surface of the solid may be curved, 
or not parallel to the XY plane, even when plane. 
The expression for the volume of the solid is 

V = J Jzrd(j>dr, (1) 

where z must be expressed in terms of r or (^ in order to effect 
the integration, and where the hmits are to be such as will 
give the volume sought. 

Corollary. — 5r/7 = zr dcp dr is a right prism with rectangu- 
lar base. The double integral in (1) is the Umit of the sum of 
the elementary solids into which the given soHd is conceived 
to be divided, or it is to be considered simply as the anti- 
differential of a second partial differential, when the differ- 
entials are taken as finite constants. Either way of regard- 
ing the differentials leads to the same result. 

Example 1. — To find the volume of a sphere of radius a. 
Taking the pole at the center of the sphere, by (1), 



J^27r fa . 
/ Va''-r^rd(f>dr 
«/o 

X27r 
d(f) = ^ ira^. 

Example 2. — A cylindrical core with h as the radius of a 
right section is cut from a sphere of radius a. Find the 



326 INTEGRAL CALCULUS 

volume of the remaining portion of the sphere, when h < a 
and the axis of the core includes a diameter of the sphere. 

7 = 2 r^ ; Va2 - r^rd(l)dr 

Jo Jb 

:. Vol. of core = ~{a^- (a" - V')i). 

O 

Example 3. — Find the volume in first o^^tant cut from a 
right cylinder, with its base of radius ri on XF plane and axis 

the 2!-axis, by the plane - + ?+-= 1. Here 

a c 

V = c j I (1 Goscj) — J- sin (I)] rd(l)dr 

^ rrn' riYcos0 , sin0\1 ,, „ Ftt n/1 , 1\-| 

Example 4. — Find the volume of Ex. 8, Exercise XXXIX, 
by using formula (1), which will give the volume sought 



more 



easily than by i / j dx dy dz. 

IT 

Va" - rH d4>dr = I {it - %) a\ 

Example 5. — Find the volume of the segment of the right 
cylinder which has its base a loop of the lemniscate r^ = 
a^ cos 2 (^ in the XY plane and its upper surface a plane 
which intersects the XY plane in the ^/-axis at an angle of 45°. 



* MASS 327 

Here z = x = rcos<}); 



Jf*^ r*a^ COS 2 4> 
I r cos 0r d<l> dr 

«/0 

= -^ I cos 2 2 cos a 

o Jo 



r- I (1 - 2sin2 0)^cos0(^ 

5 Jo 



2 
3 

7rV2a' 



16 

171. Mass. Mean Density. — As stated in Art. 154, 
the mass of a body, being defined as the product of density 
and volume, when the density * varies continuously, 

m = KmitT7A7= fydV, (1) 

AF=0 ^ J 

which becomes m = j I j ydxdydz, (2) 

/ / j yp'^sinddd dcf) dp, (3) 



or m 



according as rectangular or polar elements of volume are 
used. In these expressions y denotes the varjdng density 
at the different points within the body. The mean density 
of the body, denoted by y, is given by the equation 

fydV 

7 = y = —y- ' (4) 

When the mass is considered as distributed continuously 
over a surface, the element of volume dV is replaced by 
dA = dx dy or pdd dp; and when the mass is considered as 

* Density will now be denoted by y, instead of p used in Art. 154. 



328 INTEGRAL CALCULUS •^ 

distributed along a line, straight or curved, dV is replaced 

by dsj the element of length. 

When the integral is considered as an anti-differential the 
elements are expressed directly in terms of the finite differen- 
tials; when, however, the integral is considered as the limit 
of a sum or sums, the elements are expressed in terms of the 
infinitesimal increments, the differentials appearing under 
the integral sign. 

Example 1. — Find the mean density of a sphere in which 
the density varies as the square of the distance from the 
center. 

Here the distance p of a volume element determining its 
density, the polar element should be used. 

Taking the density at a distance p from the center as 
kp^j h being a constant, and the volume element as 
p2 sin d A0 A0 Ap, from (4) , 



n27r Pa 
I kp^8indded(l>dp 
Jo 



= ha\ 



fTra^ 5 

Again, since the density is the same for all points at the same 
distance p from the center, taking for the volume element 
a spherical shell of thickness Ap, AF = 4 Trp^ Ap, whence 

47r Ikp'dp 
T = 1 5 = H ^^ ' 

4 Trno K 



Tra^ 5 



The mean density is thus shown to be three-fifths the 
density at the surface of the sphere. 

[The mean density of the earth according to the best 
determinations is very nearly 5.527 times that of water, 
while the average density of rocks at or near the surface 
is only about two and a half times that of water; hence, 
the mean density of the earth is about twice the average 
density at the surface.] See Corollary at end of Art. 190. 



MASS 329 

Example 2, — Find the mass and mean density of a semi- 
circular plate of radius a, whose density varies as the distance 
from the bounding diameter. Here y = ky, 



m 



I kydxdy = f W; 

•a*/0 



- _ ^ka^ _ 4:ka 
I ira^ 3 TT 



Or m = I I kp^ sin Odd dp = i ka^, 

Jo Jo 

where ky = kr sin 6. 

By a single integration, the element of area being 



X • A?/ = Va^ — y'^ A^/, 
m = 2 jkVa^ - y'^ydy = f fca^. 

Example 3. — Find the mean density of a straight wire of 
length ly the density of which varies as the distance from 

one end. 

n 
ksds , , 
'0 kl 



X' 



Example 4. — Find the mass and mean density of a hemi- 
spherical solid, radius a, the density varying as the distance 
from the base. 



m 



kZTTX^ dz = I kw (a^ — z"^) zdz = j irka"^; 
Jo 



iTr/ba^ 3, 
•• ^ = 1^=8^^- 

Here the element of volume is a spherical segment, ttx^ ^.z = 
TT (a^ — z^) As!, at a distance z from the base. 

Example 5. — Find the mean density of a right circular 



330 INTEGRAL CALCULUS 

cone of height h, in which the density varies as the distance 
from a plane through the vertex perpendicular to the axis. 

/kzTTX^dz kTrjii I z^dz ^ 
Here origin is taken at the vertex and element of volume is 

irX^ Az = TTji: Z^, 

a being radius of base; 7 = kz. 



CHAPTER VI. 
MOMENT OF INERTIA. CENTER OF GRAVITY. 

172. Moment of a Force about an Axis. — The moment 
of a force about an axis perpendicular to its line of direction 
is the product of the magnitude of the force and the length 
of the perpendicular from the axis to the Une of action of 
the force. The moment is the measure of the tendency of 
the force to produce rotation about the axis. 

The moment of a force about a point is identical with the 
moment about an axis through the point, perpendicular to 
the plane containing the point and the hne of action of the 
force. 

173. First Moments. — Let a line, surface, or solid be 
divided into elements; let each element (As, A A or AF) be 
multiplied by the distance of a chosen point within the 
element from a reference line or plane. 

The limit of the sum of these products as the elements are 
taken smaller and smaller is called the first moment of the 
line, surface, or solid. 

For the first moment Mx of a plane curve about the x-axis, 

Mx = \\my.y^s= [yds; (1) 

As=0 ^ J 

and for the first moment of a plane area about the same axis, 

M, = lim T 2/ AA = fy dA. (2) 

The first moment of a solid with respect to one of the co- 
ordinate planes, say the XY plane, is given by the equation, 

M,y =\im y,z^V = fzdV. (3) 

331 



332 INTEGRAL CALCULUS 

For A A and AF appropriate expressions for the area and 
volume elements are to be used and the values corresponding 
for dA and dV substituted, in order to effect the integrations. 
The elements may be so taken that a single integration is 
sufficient, but double or triple integration will in general be 
required. 

174. Center of Gravity of a Body. — Let a given mass be 
referred to a system of rectangular coordinates, and let Mxy, 
Myz, Mxz, denote the first moments with respect to the three 
coordinate planes. 

The first moments of the mass of a soHd are derived from 
those of the geometrical sohd by the introduction of a 
density factor. 

There will be a point G {x, y, z), given by the equations, 

JyxdV fyydV JyzdV 

m^ I ydV; x=— , y=-p, -, ^ = -p y (4) 

ydV J ydV JydV 

in which the letter y denotes the density. 

The point G thus determined is the centroid of the mass. 
It is also the center of gravity of the weight W, Since W = 
mg, the masses of particles of a body are directly proportional 
to their weights; hence, the center of gravity is the same as 
the center of mass. The force of gravity acting on any mass 
is an example of a force distributed through a volume. If 
w denote the weight per unit of volume, at any point in a 
given mass, W its entire weight, and x, y, z, the coordinates 
of the center of gravity, the point where the resultant force 
exerted by gravity would act; then, from (4) or (3), 

// wxdV j wydV I i 

wdV; x=^ , y = ^ , -z = ^ 
jwdV . wdV jwdV 

If in (4) and in (5), y and w are constant, that is, if the mass 



wzdV 

-. (5) 



CENTER OF GRAVITY OF A PLANE SURFACE 333 

is homogeneous, they may be taken from under the integral 
;Sign and canceled; whence, 



fxdV JydV JzdV 



m=yV,W = wV; x=~—^^^-, y= — y— , z= y , (6) 

the coordinates of the centroid of a volume, or of a homo- 
geneous body. 

The quantities j xdV = xV, f ydV = yV, J zdV = zV, 

equal the moments of the volume with respect to the YZ, XZ, 
and XY planes, respectively. 

175. Center of Gravity of a Plane Surface. — If in the 
formulas (6) for the centroid of a volume dV is replaced by 
t dA and z taken equal to zero, then 

I xtdA j xdA j ytdA I ydA 

V = tA,x = ^ = '^ ,y=^. = ^ , (1) 

ItdA I dA tdA I dA 

where the point (x, y) is in the XY plane, and in which dA 
is the area of an element of the surface of a thin plate of 
uniform thickness and material, making tdA = dV. HA 
be the area of the middle layer, it is evident that 



xA = I xdA and yA = j ydA, 



(2) 



which are called the moments of the area A with respect to 
the i/-axis and the a;-axis. By the center of gravity of a 
plane surface is meant that point which is the center of 
gravity of a thin plate of uniform thickness and material 
whose middle layer is the surface given. The formulas for 
its coordinates are, therefore, those given in (1), 



j xdA I ydA 

^ = — A — > y = — A — 



334 INTEGRAL CALCULUS 

It is evident that the moment of an area about an axis 
through its center of gravity will be equal to zero. 

176. Center of Gravity of any Surface. — The formulas 
(6), Art. 174, become for any surface, 

j xdA I ydA I zdA 

^=^—' y-^r-' ^=-x- (i> 

By the same method as in Art. 174, it can be shown that the 
coordinates of the center of gravity of any surface, plane or 
curved, are given by the equations (1). 

177. Center of Gravity of a Line. — If in the formulas 
(5), Art. 174, for the coordinates of the center of gravity of 
a body of weight W, w dV is replaced by w ds, and ^ = 0; 
then, 

j wxds j xds j wyds I yds 

^ = -n — = -p — ' ^"'T — ^~r — ' ^^^ 

I wds j ds . j wds I ds 

where the point (x, y) is in the XY plane, and in which w ds 
is the weight of an elementary length of a slender rod of 
uniform section and material whose weight per unit of 
length is equal io w. If s be the length of the center line of 
the rod, it is evident that 

{ xds and ys = j y ds, (2) 

which are called the moments of the line with respect to the 
2/-axis and the a:-axis. The rod may be straight, in whicJi 
case the center of gravity will be at the middle point of its 
center Une. If the center line of the rod is a plane curve in 
the plane XY, the coordinates of the center of gravity are 
given by (1). By the expression, center of gravity of a line, 
is meant the point which is the center of gravity of a slender 
rod of uniform section and material, of which the given Une 



xs 



CENTER OF GRAVITY OF A SYSTEM OF BODIES 335 

is the center line. The coordinates of a Hne are, therefore, 
those given in (1). It is evident that the moment of a Hne 
about an axis through its center of gravity will be equal 
to zero. 

If the center line of the rod, or any given line, is not a plane 
curve, from (5) as before, the equations are 



j xds I yds I A 

ds Jds J 



ds 

The moments of the line with respect to the YZ, XZ, and 
XY planes, respectively, will be 



xs 



= j xds, ys = j y ds, zs = I '< 



178. Center of Gravity of a System of Bodies. — If, 

instead of a single body, there is a system of bodies whose 
volumes are Vi, Yi, Yz, . . . Yn, the coordinates of their 
centers of gravity being, respectively, (^i, yi, Zi), etc.; and, 
if (^0, yo, io) denote the coordinates of the center of gravity 
of the system and Yo its total volume, thtn 

Fo = Fi + 72 + F3 + • . • + Yn; 

Myz = YlXi + 72^2 + • • • YnXn = ^ ^= ^0^0' 

Similarly, 

M:cz=^Yy ^ Yoyo, and M^y = ^Yz= YqZq. 
Hence, 

_ X^'^ - X^y - 2^ 

where the numerators are the sum of the moments of the 
system with the respective coordinate plane, the equalities 
following from equations (6) of Art. 174. Similar equations 
hold for weights or masses upon substituting T7 or m for F, 
and for any group of surfaces by substituting A, where Aq 



336 INTEGRAL CALCULUS 

is the sum of the areas of the several surfaces. If the sur- 
faces are plane, 

The last equations are useful in getting the center of 
gravity of plane figures composed of parts, the centers of 
gravity of which are known or easily found. 

Similar equations hold for a system of lines; so being the 
sum of the lines, 

xo = ^ — , y = ^^ — , z = '^^—j 

So So So 

if the lines are not all in one plane; and ^o = when they 
are in one plane, taken as the plane of XY. 

179. The Theorems of Pappus and Guldin — First 
Theorem. — An arc of a plane curve revolving about an axis in 
the plane of the curve, hut not intersecting it, generates a surface 
of revolution, the area of which equals the product of the length 
of the revolving arc and the length of the path described by its 
center of gravity. 

Second Theorem. — A plane area, hounded by a closed 
curve, revolving about an axis in its plane but outside the area, 
generates a solid, the volume of which equals the product of the 
revolving area and the distance traveled by its center of gravity. 

To prove the first theorem; let the x-axis be the axis of 
revolution, then the surface generated by the revolution of 
the curve about the x-axis is 



./. 



yds. (1) Art. 160. 
From (2), Art. 177, for a plane curve, 

ys = j yds. 
Hence >S = 2Tys. (1) 



THE THEOREMS OF PAPPUS AND GULDIN 337 

It is evident that, if only part of a revolution is made, the 
area of the surface generated will be given by 

Si = dys, (2) 

where 6 denotes the angle in radians through which the plane 
containing the curve is turned. It is to be noted that the 
theorem and proof include the case of a segment of a straight 
line revolving about any axis. 

To prove the second theorem; let the a:-axis be the axis 
of revolution, as before; then, denoting by A A an element 
of the plane area, the volume generated by a complete 
revolution of the area is 

V = lim y,2^ry^A = 27r CydA, 

Now fydA = yA; (2) Art. 175; 

hence, V = 27ryA. (3) 

It is evident that if only part of a revolution is made, the 
angle turned through by the plane of the area being d 
radians, the volume generated will be given by 

Vi = eyA' (4) 

Example 1. — Find the center of gravity of a semicircle of 
radius a. Find it for the semicircular arc also. 

Taking the diameter along the iZ-axis, the length of the 
path described by the center of gravity as the semicircular 
area is revolved about the i/-axis is 2 irx. The semicircle by 
its revolution describes a sphere whose volume is | ira^; 
hence, by the second theorem of Pappus, 

2 TTX • i Tra^ = f ira^j 

- 4.a 



338 



INTEGRAL CALCULUS 



Also the arc describes the surface of a sphere, 4^a^; hence, 
by the first theorem of Pappus, 

27rX"ira = 4:7ra^; 

_ 2a 
.'. X = 



Example 2. — Find the center of gravity of the semi- 

X 77 

elUpse, - + ^ = 1. 

Taking the x-axis as the axis of symmetry and applying 

the second theorem of Pappus 
as in Ex. 1, it is found that 

X = r— , also ; hence, as ^ = 0, 

OTT 

the centers of gravity of the 
two areas are identical. 

Example 3. — Find the 
center of gravity of the 
'^ ^ quadrant of the ellipse. 

Let AOB be the quadrant 
of the elUpse, and the element of area a narrow strip parallel 
to ?/-axis. 

dA = ydx =^ - Va^ — x^ dx. 
a 

/xdA - I X Va^ — x^ dx 
a Jo 




X = 



Iwah 



Ih^^'-^'H 



jirab 



_ 4 a /The same abscissaN 
3 TT \as for semi-ellipse . / 



-46 
Similarly, it is found that 2/ = 5— • 

OTT 

_ _ 4a 
Corollary. — For the circle, x^ -\-y^ = a^; a; = 2/ = 5 — 



EXAMPLES OF CENTER OF GRAVITY 



339 



/ 



Example 4. — Find the center of gravity of a circular arc. 
Let AB be a circular arc, whose radius is a and whose center 
is at origin 0. Let Oi and 02 be 
the angles with a;-axis made by 
the radii OB and OA to the ends 
of the arc. From 

xds I yds 
X = —^ and y = '-p ; 

I ds I ds 

using polar coordinates, x = a 
cos d, y = a sin 6, ds = a dO; 



o? cos QdB a sin / . ^ - ^\ 
J^ _ a (sm ^1 — sm 02) 




and 



/ a^si 

— *J 09 



sin 6 dd 






-\9. 
— acos0 , - .. 
J92 _ g (cos 62 — cos Bi) 

J92 



di — 02 



When 02 = 0, the equations reduce to 



X = 



a sin 01 
01 



y = 



a(l — cos^i) 
01 



Corollary.— When 0i = ^1, ^2 = -^1; x -- ^^^^ ^ 



When 



01 ' 
01 = 90°, 02 = 0; x = — , j^=— - 



0. 



Example 5. — Find the center of gravity of a triangle. 
Let the triangle ABC have base h and altitude h, and let the 
rc-axis be through the vertex parallel to the base, and the 
2/-axis positive downwards. Take dA = L dy, where L is 



340 



INTEGRAL CALCULUS 



the length of an elementary strip, parallel to the base and at 
a distance y from x-axis. 



dA = j^ydy; 



hence 



L :y = h :h; 



y = 






Similarly, by taking strips parallel to h or the 2/-axis, 



X = 



-h 








A 






a: 


1 / 
\ /l 

VI 










l„y- 


///////F/M 




B 
Y 


/I 

/ ! 




m 


C 





Hence the perpendicular distance of the center of gravity 
from the base will be \ h, and, since the center of gravity of 
each elementary strip will be its middle point, the center of 
gravity of the triangle will be on the median line Am, at 
one- third the distance from m to A. Similarly, it is on the 
median line from B; hence it is at the intersection of the 
medians. 

Example 6. — Find the center of gravity of a semicircular 
plate of radius a, whose density varies as the distance from 
the center. 

Here, the density being determined by the distance from 
the center, the polar element is used. 



EXAMPLES OF CENTER OF GRAVITY 341 

Let the density 7 = kp, and the x-axis as in Ex. 1 ; then 



JyxdA f_^ fjkf^ COS ede dp 



I 



ydA 



I Ikp^dddp 



cos Odd 



27r' 



and y =" 0. 



dd 



Example 7. — Find the center 
of gravity of the volume cut from 
a right cyhnder, the radius of 
whose base is a, by the planes 
z = and z = mx, the volume 
above the plane of XF alone con- 
sidered. Here 

h 

I xdxdydz 



2h r< 

a Jo 



a2-x2 

x^Va^ 




x^dx = 



1T0?h 



X = 



Tra%/8 Tva^h/S 



W' 



F = f a^/i from Ex. 2, Exercise XXXIX. 

h 

I I zdxdydz 

./_\/a2-x2 t/o 

= -5 / x^Va^ — x^dx = — TTT-. ) 
a^ Jo 16 

- 7ra%yiQ -KamiXfS 3 , , _ 

. = — ^r- = -T^— 32^/^; and y 



0. 



342 INTEGRAL CALCULUS 

EXERCISE XL. 

Find the coordinates of the center of gravity: 

1. Area of the parabolic half segment, y'^ = 4 ax, x = to x = a. 

Ans. (f a, fa). 

2. The area under one arch of the cycloid, x = a {d — sind), 
y = a {1 — COS0). Ans. {ira, f a). 

(X _x\ 

ea -J- e aj,x = 0tox = a. 



Ans (^^ g (e^ + 4 - e-^) \ 



4. A semicircular plate of radius a, the density varying as the 
distance from the bounding diameter. Ans. {j\Tra,0). 

5. A homogeneous right circular cone, radius of base, a, and altitude 
h. The axis on the x-axis. Ans. x = lh,y=z = 0. 

6. A homogeneous paraboloid of revolution from the origin to x = h, 

Ans. X = ^h, y = 0. 

7. A hemisphere whose density varies as the distance from the base 
whose radius is a. Ans. {0, 0, ^j a.) 

8. The eighth part of a sphere in the first octant, the density of the 
mass varying as the distance from the pole or origin at the center. 

Ans. X = y = I a. 

9. The circular sector subtending the angle di, radius a. 

/ \ T? n a r \ A /2a(sin0i) 2a(l — cos0i)\ 
(a) For e,=e,. (a) Ans. (^g — ^— , 3 '^ 

_ 4 ^ 

(&) For di = 90°, the quadrant. (6) Ans. x = y = -^ — 

O TT 

(c) For di = 180°, the semicircle. (c) Ans. a; = 5— , 2/ = 0. 

O TT 

10. Find the center of gravity of a T-section. Using the method of 
Art. 178, with the dimensions on figure, taking the moments of the three 
rectangles composing the section, 

y, _S^^_ 4X7| + 6X4 + 8X| _ 58 _ 29 _ 

y' Ao 4 + 6 + 8 "18-9 -^-2291118. 

xo = 0, as the center of gravity will be on the axis of symmetry. Here 
advantage is taken of the knowledge that the center of gravity of each 
of the rectangles is at the center of the rectangle. 

Note. — This is the method of finding the centers of gravity of the 
various shapes, or built-up sections, used in constructions. 



EXAMPLES OF CENTER OF GRAVITY 



343 



J- 



k-/l<'> 



I/O 



\o 



11. Find the center of gravity of the trapezoid OAao. Let the upper 
base be 6 and the lower base B, the altitude h. Divide the trapezoid 
into two triangles by diagonal Oa, then the distances of the centers of 
gravity from OX are \ h and | h for the triangles, respectively. Then 
Bh h hh 2h 
2 ^3'^2^ 3 ^h /B_±2h\ 
3\B + bJ' 



2/0 = 



X^y 



(B+h) 



t 



V^rC 



-^ 

m 

^"TX ~ 






y 






M 
-B- 




The center of gravity of each strip of area parallel to the bases will be 
its middle point, hence the center of gravity of the whole area is on the 
median line mM of the trapezoid. Graphically, the center of gravity 
is located by again dividing the trapezoid into two other triangles by 
the diagonal oA, and drawing lines connecting each pair of centers of 
gravity; the intersection of these connecting lines is the center of 
gravity G of the trapezoid. 



344 INTEGRAL CALCULUS 

12. From Art. 178, show that the center of gravity of two volumes, 
masses, areas, or Hnes, lies on the line joining their separate centers of 
gravity and divides that line into segments inversely proportional to 
the two magnitudes. 

In Ex. 11, the point G, the center of gravity of the trapezoid divides 
the line GiG-i connecting the centers of gravity of two triangles, inversely 
as the areas of the triangles, and it divides the line G'G" in the same 
way. 

In this way the common center of gravity of the Earth and the Sun 
is found. Taking the distance from the Earth to the Sun as 92,400,000 
miles, and the mass of the Sun as 327,000 times that of the Earth, 
makes the distance of the common center of gravity from the center of 

the Sun ' ' miles, or only about 280 miles; so that the Sun is 

considered practically at rest relative to the Earth. 

180. Second Moments — Moment of Inertia. — The 

term moment of inertia is applied to a number of expressions 
which are second moments of lines, of areas, or of solids. 

Let each of the elements of length, of area, or of volume 
(As, AA, or Ay), into which a line, a surface, or a soUd may- 
be supposed to be divided, be multiplied by the square of the 
distance of some chosen point in the element from a reference 
line or plane. 

The hmit of the sum of these products as the elements are 
taken smaller and smaller is called the second moment of the 
given line, surface, or solid with respect to the reference line 
or plane. 

Formulas for second moments are derived from those for 
first moments by squaring the distance factor. Denoting 
the second moment by I, the general symbol for moment of 
inertia, the following formulas correspond to (1), (2), (3), 
Art. 173. 

(1) 
(2) 
(3) 



For a plane curve, 


h = Jy'ds. 


For a plane area, 


h = fy'dA 


For a volume, 


hy = Jz-'dV. 



POLAR MOMENT OF INERTIA 345 

As applied to an area, the moment of inertia is a numerical 
quantity entering into a large number of engineering com- 
putations and takes its name from the analogy between the 
mathematical expression for it and that for the moment of 
inertia of a mass or solid. It is evident from the form of the 
expression that the moment of inertia is always a positive 
quantity, being unhke the first moment in that respect. In 
distinction from the moment of inertia, the first moment is 
sometimes termed the statical moment. 

181. Radius of Gyration. — The radius of gyration of a 
solid is the distance from the reference line, called the in- 
ertia axis, to that point in the solid at which, if its entire 
mass could be concentrated, its moment of inertia would be 
unchanged. Thus, if m, the entire mass of a body, be con- 
sidered as concentrated at a point, and k denote the distance 
from the inertia axis to that point, the expression for the 

moment of inertia, / r^ dm, will be equal to k^m. Therefore, 

/ = k^m and k = \ ~, k being the radius of gyration of 

the solid of mass m. 

In the case of the plane area, by analogy, 

h = fx^dA = k^A, and k = \/j, 

where A is the entire area and k is its radius of gyration with 
respect to the x-axis. 

The radius of gyration of both the soUd mass and the 
plane area will evidently be expressed in hnear units. 

182. Polar Moment of Inertia. — The polar moment of 
inertia of a plane area is the moment of inertia about an axis 
perpendicular to its plane. The moments of inertia about 
any two rectangular axes in the plane of the area are called 
rectangular moments of inertia when they are mentioned 
in connection with the polar moment. It is evident that 



346 



INTEGRAL CALCULUS 



for an area in the plane of XF, the moment of inertia about 
the ^-axis is a polar moment of inertia, and that it is equal 
to the sum of the two rectangular moments. 



y 

'0 



Thus, let the point F (x, y, 0) be in the element of area, 

then, L= jr'^dA 

is the polar moment of inertia of the area about the 2;-axis or 
with respect to the point in its plane. But since 



x' + y\ L = J{y' + x')dA, 



hence, 1^ = 1^ + ly. 

The symbol for the polar moment of inertia is, in general, 
the letter J. 

183. Moments of Inertia about Parallel Axes. — 

Theorem. — The moment of inertia of an area about an axis 
in its plane, not passing through its center of gravity, is equal 
to its moment of inertia about a parallel axis, passing through 
its center of gravity, increased by the product of the area and 
the square of the distance between the two axes. 



MOMENTS OF INERTIA ABOUT PARALLEL AXES 347 

Let the origin be at Og, the center of gravity of the area, 
and take the i/-axis parallel to the inertia axis through in 
the plane. Let the point P {x, y) be in the element of area, 




then its coordinates with respect to the axis through are 
(x + a, y). The moment of inertia Ig= j x^ dA, and that 
about the aids through is 

I^ C{x + aydA 
= f{x' + a'-{-2ax)dA 
= fx''dA+ fa''dA+ C2axdA 
= 7G-f Aa2 + 2a CxdA. (1) 

The quantity I x dA must be equal to zero, since the 2/-axis 
passes through the center of gravity (Art. 175). Therefore, 

I = lG + Aa\ (2) 

It is evident that equation (1) gives the relation between 
the moments of inertia with respect to any two parallel axes 
in the plane of the area, OgY being replaced by an axis 
through any point in the plane. 

In the same way it can be proved that the polar moment 
of inertia of the area, with respect to any point 0, is equal 
to its polar moment of inertia with respect to its center of 



348 INTEGRAL CALCULUS 

gravity plus the product of the area and the square of the 
distance between the point and the center of gravity. 

Corollary 1. — It follows from (2), that, of all parallel axes, 
the axis through the center of gravity, called the gravity 
axis, has the least moment of inertia. 

Corollary 2. — When the inertia axis is a gravity axis, the 
radius of gyration, then called the principal radius of gyra- 
tion, is the least radius for parallel axes; from (2), 

AF = Aka^ + Aa^ :. k^ = kg^ + a^, (3) 

where kg . is the principal radius of gyration and a is the 
distance between it and a parallel axis. 

184. Product of Inertia of a Plane Area. — The product 
of inertia of a plane area is a numerical quantity which is of 
value only as it is found to enter into the determination of 
the relations between moments of inertia with respect to 
different axes. The product of inertia of an area A with 
respect to the axes of x and y is a, second moment, 



/ 



xydA, 



and may be defined as the limit of the sum of the products 
of the elementary areas and the product of their distances 
from the two coordinate axes. Unlike the moment of 
inertia, the product of inertia may evidently be either posi- 
tive or negative, depending upon its distribution in the 
different quadrants; and the area may be so located that 
its product of inertia will be zero. 

The axes may be so chosen as to make the product of 
inertia of an area zero, and such axes are called principal 
axes, the corresponding moments of inertia being called 
principal moments of inertia. It can be shown that for any 
point of an area (or body) there exists a pair of rectangular 

axes for which I xy dA = 0, and that the moment of inertia 

is a maximum when taken with respect to one of the principal 



LEAST MOMENT OF INERTIA 349 

axes, and a minimum when taken with respect to the other. 
The relation between the polar moment and the two rec- 
tangular moments (Art. 182) shows that if one of the two is 
a maximum the other is a minimum, and vice versa. 

185. Least Moment of Inertia. — In designing a column 
an engineer needs to know the least radius of gyration, and 
consequently the least moment of inertia, of the cross section, 
since the resistance to bending is least about that axis which 
has the least moment of inertia. It has been stated that 
the moment of inertia of an area is least about a principal 
axis through the center of gravity, and it is necessary, there- 
fore, to determine those principal axes which pass through 
the center of gravity. 

In many cases the position of the principal axes are known 

at once, for all axes of symmetry are principal axes, / xy dA 

being equal to zero for such axes, since for every point {x, y) 
there is another (x, —y). 

For example, any diameter of a circular area, the axis of 
a parabola, either axis of the ellipse or of the hyperbola is 
a principal axis. For a rectangle it is obvious that the lines 
through the center parallel to the sides are principal axes, 
but the diagonal of a rectangular plate is not a principal 
axis at its middle point. The gravity axes parallel and 
perpendicular to the web of the cross section of an /-beam, 
channel, or T-beam are principal axes. 

When the section is unsymmetrical, it is necessary to 

evaluate the integral, I xy dA, in determining the principal 

axes. 

Remark. — A full treatment of the subject of moment of 
inertia and product of inertia is beyond the scope of this 
book. What has been given has been confined for the most 
part to areas, as that part of the subject has more immediate 
application in engineering. 



350 



INTEGRAL CALCULUS 



186. Deduction of Formulas for Moment of Inertia. — 

1. Rectangle of base h and altitude h: 



L = fy' dA = J'bf dy = tV W. 

§)■=! 



la'b' =^h-\-Aa'^j^ h¥ + bh 



hh\ 



ly = jx^dA = r hx^dx = j\¥h. 



bh 



Again, 



Ia'b' = £by'dy = ibh\ 
Ix = tV^^ J = |6^; for square. 



A 






Y 


f 


















X'-^- 
A' 


-- 


- 


_i__. 


X 

B' 









2. Triangle about the axes: 

(a) Through the apex parallel to the base. 

(5) Through the center of gravity parallel to the 

(c) Through the base. 



(1) 

(2) 
(3) 



J = h + h = f^hh' + ^Vh = '^{V + h^). (4) 



(2') 



DEDUCTION OF FORMULAS 
Let the base be b and the altitude h. 



,. J r . . , b¥ .bh /h^\ hW 










y 


1 

1 


/ 


\ 




1 


/////////a 


y//m. 




1 
h 


/ 




V 




\ 


- \....... 


b- 


/ 



3. Circle: 

(a) Polar moment of inertia, axis through center. 
(6) Moment of inertia about a diameter. 




(a) J = Tr^ dA = J\ tt/ dp = ^' 



wd^ 
32* 



351 

(1) 
(2) 
(3) 



(1) 



352 INTEGRAL CALCULUS 

For a sector with angle 6, 

J = j'dpdp = 



4' 



_ ^1 _ 7rr^ 



(2) 
(3) 

(4) 



/. + /, = 27, = J 
For a circular quadrant, 7, = 7 
For a semicircle, Ix — I 



(6) Since for the circle the moments of inertia about all 
diameters are equal, 

16' 

_ _ _ 7rr* 
^~2^ ~ 8 * 

4. Ellipse: 



_ 1 

^"2 

1 



(5) 
(6) 
(7) 



7, = Jx' dA = ^/"x2 (a2 - a;2)l d(a; = ^. 



/x + 7, = ^(a2 + 62). 



(1) 
(2) 
(3) 




187. Moment of Inertia of Compound Areas. — Since 
the moment of inertia is always positive, the moment of 
inertia of an area about any axis is equal to the sum of the 



MOMENT OF INERTIA OF COMPOUND AREAS 353 

moments of inertia, about that axis, of the parts into which 
the area may be divided. In some cases the area being 
considered the difference of two areas, its moment of inertia 
will be equal to the difference of the moments of inertia of 
the two areas. 

Example 1 . — Find the moment 
of inertia of the T-section shown 
with the dimensions on figure. 

(a) About the axis of X 
through top of section. 

(6) About the axis Zo through 
the center of gravity. 

(a) 7. = i6¥ = 1(4- 1) P + i (1 X 4^) = !+¥ = ¥- (ins.)*. 



4" V 

-x-| 1 1 1 — a: — ^ 


f" 




■^ 


</ 


1" 





(h) yo = 



^Ay 



3X i 



+ 4X2 



9i 19. 
-^ = Ti ins. 
7 14 



A 3 + 4 

/o = /x-At/o2 = -V—7(H)' = 22.33-9.32= 13.01 (ins.)*. 

Example 2. — Hollow rectangular sec- 
tion: 

/. = tV (b¥ - hihi'), tV Q>' - h") for 
hollow square. 



h 



Example 3. — Hollow circular section: 



/ = T (^2* — n^), about a diameter; 



J = 


2 {t2^ — Ti^) , about axis 
center. 


through 


Or in 


terms of the diameters: 




/ = 


^(^^-^1^), 




J = 


l^W-d.^), 






354 



INTEGRAL CALCULUS 



Example 4. — Find the moment of inertia of the section 
shown. 



3 



= ^ (6)^ - ^' - T (2)^ (3)^ + 2.3-4^ = 46.33 (ins.)*. 
J = h + Iy = 92.66 (ins.)*. 




Example 5. — Find the moment of inertia of the trapezoid: 
(a) about its lower base; (b) about the gravity axis. 

ix ^on -t^2^fi 4 ^ 12 

= 216 + 144 = 360 (ins.)^. 
Ig = Ix- Aa' = 360 - 36 (f )2 = 104 (ins.)*. 




CHAPTER VII. 

APPLICATIONS. PRESSURE. STRESS. 
ATTRACTION. 

188. Intensity of a Distributed Force. — A distributed 
force is one that acts on a surface, such as the pressure of 
water against the surface of contact, the pressure of a weight 
upon the surface of its support; or, one that acts through a 
given volume, such as the attraction of the earth on a body. 

All forces are really distributed forces since no finite force 
can act at a point of no area; although this is true, in some 
cases it is convenient to regard a force, whose place of appli- 
cation is small, as though it were applied at a point. Such 
a force is called a concentrated force. A distributed force is 
conceived as ''equivalent to" a concentrated force called 
the resultant force, when the force of gravity acting on every 
particle of a body is taken as acting at a point within the 
body, called the center of gravity. A distributed force is 
regarded as the limiting case of a system of concentrated 
forces whose number becomes larger as their individual 
magnitudes become smaller. It is thus that a force is re- 
garded as having a definite point of application and a definite 
line of action: when so regarded it is a localized vector quantity. 
When a force is distributed over an area, the intensity of the 
force at a point is the number of units of force acting on a 
unit of area including that point. 

Briefly the intensity is defined as the force per unit of 
area. If the force is uniformly distributed, the intensity p 
will be equal to the force P, acting on the entire area, divided 
by the area A ; that is, 

.-f- (1) 

355 



356 INTEGRAL CALCULUS 

If the force is not uniformly distributed, the intensity at 
any point of the area will be given by 

r fAPl dP 

the hmit of the ratio of the force, acting on a small element 
of the area, to that element as it approaches zero as a hmit. 
When the intensity varies from point to point over any area, 
the force on that area divided by that area gives the average 
intensity on the area. In any case the entire force is given 
by 



P 



= JpdA, (3) 



where p, if variable, must be expressed in the same terms as 
dA in order to get P by integration. 
If p is constant. 



V 



j dA = pAy and 2^ = T* (1) 



189. Pressure of Liquids. — The pressure of a liquid 
on a surface is normal to the surface, and the intensity of 
pressure varies as the depth of the point below the free 
surface of the hquid. The intensity is given by 

p = wh, (1) 

where w is the weight of a cubic unit of the liquid and h, 
called the head, is the depth of the point below the free 
surface. 

If w is expressed in pounds per cubic foot, h should be in 
feet, and p will then result in pounds per square foot. For 
water w is usually taken as 62J lbs. per cubic foot, and the 
intensity of pressure given in pounds per square foot. When 
the intensity p is constant on any horizontally immersed 
plane surface the total pressure P is, by (1) Art. 188, 

P = Awh = Q2.b Ahlh^. (2) 



PRESSURE OF LIQUIDS 357 

When the surface under pressure is not horizontal, by (3), 
Art. 188, 



hi 



wxdAf (3) 



where the Umits of x are the least and greatest heads on the 
area. When the area extends to the surface of the liquid, 
the lower limit becomes zero and the upper may be taken 
as h. 

Since in (3), j xdA =xA, by (2), Art. 175, 



=/ 



wxdA = Awx = 62.5 Ax lbs. 



Hence, the total pressure on an immersed area is the product of 
that area, the weight of a cubic unit of water, and the head upon 
its center of gravity. 

In general, the pressure of any liquid upon an area is equal 
to the weight of a column of liquid whose base is the area pressed 
and whose height is the depth of the center of gravity of the area 
helow the surface. 

Example 1. — The vertical face of a dam subjected to the 
pressure of water is h ft. in height and b ft. in breadth. The 
pressure of the water varies as the depth; the intensity at 
a depth x is wx, w being the constant weight of a cubic unit 
of water. Required the total pressure on the face of the 
dam, and the location of the center of pressure. 

Let the area of pressure be divided into strips of width Ax 
and length b, then wx -b /S.x is approximately the pressure on 
the element of area — for wx is the intensity of pressure at 
the top of the strip. 

The sum of a finite number of terms of the form wbx Ax 
would give a result for the total pressure less than the actual 
value; but the exact value is 

_. ,. ^'^ , . 7 P J wbx^l'' wbh'^ wh ,, .^. 

P= lim X wbx/!:ix = wb I xdx= —^^ =—^r-=-^'bh. (1) 



358 



INTEGRAL CALCULUS 



The intensity of pressure is a uniformly varying force having 
zero value at the surface of the water and value wh at the 
bottom. The center of pressure, being the point of applica- 
tion of the resultant pressure, is given by taking the moment 



^=^A 




of P about the surface line equal to the limit of the sum of 
the moments of the elementary pressures about that line : 

X'P = I whx^ ax = —5— = —5— , 
Jo o Jo o 



£ 



whx'^ dx 



i 



whxdx 



wb¥/S 

wbhy2 



h- 



(2) 



In general, the center of pressure of a rectangle with a 
side at the surface is two-thirds the height of the rectangle 
below the surface. When the top of the area is hi below the 
surface and the bottom is h below, the total pressure is 

tJhi ^ 

and 

M x^' 
2 



h 

hi 



x^dx 



2hlj-hl 
3 hi" - h/ 



(3) 



PRESSURE OF LIQUIDS 



359 



It may be noted that the second moment in the numerator 
is the moment of inertia of the area, and the first moment in 
the denominator is the statical moment. 

Note. — That P = l wbx dx in (1) is the reversal of a 



rate may be seen by considering the rate of change of the 
total pressure when the depth x is increased by Ax, for then 
the pressure P on the area is increased by AP= vox • h ^x, 
approximately, and AP/Ax= whx (nearly). 



Hence 



,. AP dP 

lim -T — = -7- = whx, 
Ax=o Ax dx 



the rate of change of P; and P = I whx dx as in (1); so 

the total pressure is a function of h and its rate of change 
is whx = ph, where p = wx is the pressure on a unit of 
area. 

Note. — Whenever an external force acts on a body it 
induces a resisting force within the body. This is in accord- 
ance with Newton's third 
law of motion. This in- 
ternal resistance is due to 
the molecular forces or 
stresses within the body. 
A stress is a distributed 
force acting on a surface 

Example 2. — A verti- 
cal rectangular section 
ABBiAi of a beam of 
breadth h and depth d is 
subjected to a stress of 
tension and compression uniformly varying in intensity from 
zero at the middle fiber to St and Sc at the outside fibers, at 
the distance, yi = i d, from the neutral axis or middle fiber 
of the section. 




360 INTEGRAL CALCULUS 

Find the total tensile and compressive stresses and the 
centers of stress on each half of the section. 

The intensity of stress at the distance y from the neutral 
axis is S/yiy, hence S/yiyh Ay is approximately the stress on a 
strip A 2/ in depth — the intensity at the edge of the strip 
being taken. The sum of a finite number of such terms 
would give a result less than the total stress on the half 
section; but the exact value is given by 

P.= hm2^ -ybAy=—l ydy 

Ay=0 ^0 2/1 2/1 t/0 

d 

Sh yn^^ Sh f Shd 

For the center of stress, the point of apphcation of the total 
stress, 

- D P'^^ 2^ Shy^y^ Sh J Sbd\ 
- Shd' /Shd 1 , 

When S = St = Sc, P = Pi; hence, the total tensile and 
compressive stresses form a couple with arm f d, the moment 
being 

^.U^'-^,c.neAti.esectionmodulus. 
4 o o 

By the mechanics of beams, if M denote the moment of the 

external forces acting on the beam, M = —^ — 

Example 3. — A vertical circular section of a beam is 
subjected to a stress of tension and compression uniformly 
varying in intensity from zero at the horizontal diameter 
2 a of the section to St and Sc at the top and bottom fibers, 
respectively. 



PRESSURE OF LIQUIDS 



361 



Find the total stresses on the upper and lower semicircles 
and the centers of stress on the semicircles. 




Denoting by P the total stress on either semicircle, and 
taking St = Sc = S; 

P = lim 5j -y '2x^y = — I Va^ — y'^ydy 
A2^=o "^0 o, a Jo 

y>P=—-j Va^-y^yHy 
a Jo 

= 2^r^(2.2_^2)V^J3^+^%in-i^TP^^^-2'Exer-\ 



2S TTO^ 

a ' 16 



Swa' Sird' 



8 



64 



" ^ 64/6 32 * 

Hence the couple formed by the forces P has an arm, 
2^ = TQT^d^ and 

M = -^Sd"^ • Y^Trd = ~o^j the section modulus. 

Example 4. — Find the total water pressure upon the end 
of a circular right cylinder immersed lengthwise, one element 
of the cylinder just at the surface of the water. Find the 
center of pressure of the circular area. 



362 



INTEGRAL CALCULUS 



The intensity of pressure at a depth y being wy^ 
approximate pressure on a strip iswy'2x ^y. 




the 




2wxy ^y) 

ny=v 

P = 2w f " {2ay-y'^)^ydy = 2wa ( \2ay-y^)^dy 
Jo Jo 



7ra^ 



= 2wa'-— = wira^; (Ex. 13, Exercise XXV.) 

2w I (2ay-y^)y^dy ia'2w I {2ay-y^)^ydy 

— _ Jo Jo 

^ ~ P ~ wira^ 

= i^^ =^a = ^d. (Ex. 13, Exercise XXV.) 



WTTO' 



8 



EXERCISE XLI. 

1. (6) The pressure upon one side of the gate of a dry dock, the 
wetted area being a rectangle 80 ft. long and 30 ft. deep, is to be found 
exactly. Take w = 62^ lb. for the weight of a cubic foot of water. 

(c) Find the depth of the center of pressure. Ans. (c) 20 ft. 

(a) Find the pressure approximately by a limited number of terms. 



(See Art. 154.) 




Ans. (h) 112^ tons. 

2. The pressure on the gate that closes 
a water main half full of water, the diam- 
eter of the main being 8 ft. Get the 
exact (6) pressure only, (c) Find the 
center of pressure. 

Ans.{b)P = ''i^w\ha. (c) ^ = f tt ft. 

3. Find the exact pressure on a cir- 
cular disk 10 ft. in diameter, submerged 
below water with its plane vertical and 
its center 10 ft. below the surface. Here 



ATTRACTION. LAW OF GRAVITATION 



363 



p^y^C^" {lO-y)2xdy = 2wC " {10 - y) {a^ - y^^^ dy 

J—5=—a •/— 5 =— o 

= 20 wC {d? - 2/2)^ dy -2wC (a^ - y^)^ y dy 
= 20i/; • ^/ TT + f (a2 - ?/2)^T"" = 250Trw 

J— 5=— a 

= 2507r-62|lb. 

4. Find the pressure on the face of a temporary bulkhead 4 ft. in 
diameter closing an unfinished water main, when water is let in from 
the reservoir. The center of bulkhead is 40 ft. below the surface of the 
water in the reservoir. Ans. Nearly 16 tons. 

5. Find the pressure on ,the end of a parabolic trough when it is full 
of water. The parabola has its vertex downward, the latus rectum is 
in the surface and is 4 ft. long. Here 



P = lim 2I 2w{l — y) xAy 

Ay=0 ^^0 



2w r2(l - 

iwC y'dy - C y^dy 

Jo Jo 

[1 2/^ -f2/^J^ = 11^ = 661 lbs. 



y^dy 



4 w; I ?/^ 



< 

T 


Y 
4- 




-"1 ^ 


X 


J-y 


y\ 


^y^ 


y//////////jy///////y 


'///^ / 


"^-^ 


— ^ — ^x 



6. A horizontal cylindrical tank is half full of oil weighing 50 lb. per 
cubic foot. The diameter of each end is 4 ft. Find the pressure on 
each end. Find the pressure when the tank is full also. 

Am. 2661 lb.; 12561b. 

190. * Attraction. Law of Gravitation. — Every portion 
of matter acts on every other portion of matter with forces 
of attraction or repulsion. According to Newton's Law of 
Universal Gravitation, every particle of matter attracts 
every other such particle with a force which acts along the 
line joining the two particles, and whose magnitude is pro- 

* This article is based on a discussion in Fuller and Johnston's 
Applied Mechanics. 



364 INTEGRAL CALCULUS 

portional directly to the product of their masses and in- 
versely to the square of the distance between them. 

If the masses of the particles are m and mi and the distance 
between them is r, the law may be expressed algebraically by 

F = K^, (1) 

where F is the attractive force between the particles and 
K is a constant, determined by experiment, its numerical 
value depending on the units in which F, m, mi and r are 
expressed. The value of K having been determined in one 
case is then known for all cases. 

While formula (1) expresses the law of gravitation, the 
general algebraic expression for the law of attraction would 
be 

where 0(r) is some function of the distance between the 
particles, depending on the nature of the attractive force, 
X is a constant, and m and mi other quantities than the 
masses of particles. 

In interpreting formula (1), it is to be noted that it applies 
strictly only to particles; for the particles having finite 
masses must have finite dimensions and hence, as the distance 
between them is diminished, r cannot be less than a certain 
finite quantity and the maximum value of F, when the 
particles are in contact, will b^ a finite quantity. If r were 
taken to be zero in any case, F for finite values of m and mi 
would become oo which would be impossible under the 
conditions. 

The formula, while applying strictly only to particles, 
gives, to a close approximation, the attraction between two 
bodies of finite size, whose linear dimensions are small 
compared to the distance between them. In the appHcation 
of the law the attraction of one particle on another may be 



ATTRACTION. LAW OF GRAVITATION 365 

regarded as acting at a point. It will be shown that any 
sphere attracts any outside particle as if the whole attraction 
was towards a point at the center of the sphere, but, in 
general, the attraction of bodies on exterior particles is not 
always towards the center of gravity of the attracting body. 

Attraction of gravitation is a mutual action between two 
particles or bodies; that is, each exerts an attractive force 
upon the other, the two forces being equal in magnitude and 
opposite in direction. This is implied in the Law, and it is 
also in accordance with the law of '^ action and reaction," 
Newton's third law of motion. 

It is evident that, in formula (1), K is equal to the force 
with which two particles of unit mass at a unit distance 
apart attract each other. 

If the equation is divided by mi, then 

^=a = K^, (3) 

where a is the acceleration which would be produced in the 
mass Wi by the attraction of the mass m at a distance r. 

771 

The quantity K -^ would also equal the force of attraction 

exerted by the mass m on a mass unity at a distance r. 
Briefly this is called the attraction at the point, at which the 
unit mass is situated, exerted by the mass m. 

The attraction at a point exerted by any mass is called the 
strength of the field of force, or briefly, the strength of field, by 
which the space through which the attraction of the mass is 
exerted is expressed. 

Electrostatic and magnetic attraction and repulsion are 
other examples of forces, which are governed by laws similar 
to that of gravitation. 

The following examples are based on the law 



366 



INTEGRAL CALCULUS 



■'V 



A4^^ 






p' 



0K~ 



^-^OTp.^- 






where F is the attraction of a particle of mass m for a par- 
ticle of unit mass, the body being taken as homogeneous, of 

uniform density; that is, each 
cubic unit having the same 
weight. 

Example 1. — Attraction of a 
Rod of Uniform Section. — (a) 
Let the rod of small section be 
in the form of a circular arc; 
to find the attraction at the 
center of the circle. 

Let r be the radius, a the 
angle subtended at the center, 
and m the mass of a unit length 
of the rod. Take the axis OX 
bisecting the angle a, and let d 
be the angle which the radius 
to any point P makes with OX. The attraction at of a 
particle at P is 



N 




jB' 



AF = 



Km As Km Ad 



(1) 



Since all the elementary forces of attraction are directed 
to the point 0, the resultant R is found from the sum of the 
components of the elementary forces. 



X^=^/ 



. ,. 2 Km . a 

cos Odd = sm ^j 

r 2 



at 



~ 2 

the attractions being neutralized. 

Hence, R = ^JXW+^ljf = 

1171. D 2 Km 
When a = IT, R = 



2Km . a 

sm-- 

r 2 



(2) 
(3) 



ATTRACTION. LAW OF GRAVITATION 367 

When a = 2Tr, R = 0; since the arc being a circumfer- 
ence of a circle, the attractions neutrahze each other. 

(6) Let the rod be straight; to find the attraction at a 
point. Let r be the shortest distance from a point to the 
rod. Taking as origin, the equation of the rod is x = r 
(constant). 

When the rod is B'AB the angles may be taken as in (a) 
for the circular arc. The attraction at of a particle at P' 
on the rod is 

^^ = JOPT ^ ^ r' ^' ^ ^ 

The resultant attraction is found as in (a) ; since y = r tan 0, 

-, rdd 

2 Km .a - , ^ 

= sm-, as m (a). 

r 2 

a 

^Y = ^ J cos' d sine dy = ^T ^^^^^^ = ^' 
Hence, R = sin-, as in (a). (2') 

T A 

It is thus shown that if the straight rod B'AB is of the 
same mass per unit length as P2P1, the resultant attraction 
of B'AB at is the same as the attraction of P2-P1, since the 
sum of the attractions of the elementary masses m l^y and 
the sum of the attractions of the elementary masses m As 
have the same limit. 

(50 Let the rod be still straight but AiB, the angles with 
OX of the lines from to the ends being ai and a^] then, 

X^ Km r«2 Km , . . . 

A = I cos Odd = (sm 0:2 — sm ai), 

V^ ^^ Km r"2 . Km , . 

X F = I sm QdQ = (cos ai — cos 012). 



368 INTEGRAL CALCULUS 

Hence, 

R = V2 [1 — cos (a2 — Oil)] = Sin — ^ — -, (3') 



r 
and 

. ^ -^ ^ cos Q!2 — COS Q!i ^ Q!2 + «1 

tan 0r = S — = -• -• = tan — - — , 

V X sin a2 — sm ai 2 

0:2 + 0:1 

.. er-—^—, 

the line of action of R bisecting the angle AiOB^ subtended 
by AiB at 0. 

(6") Let the point be at A, making r = and (3') 
indeterminate. Then, 



Jy^ y^ \yi 2/2/ 



Kmiy2-yi) ^KM 
2/22/1 2/22/1' 

where M = the entire mass of the rod. 

If the point is taken at the end of the rod, 2/1 = and 
equation (4) gives R = 00 . This is impossible; for, as 
stated in Art. 190, r cannot be zero for finite particles. If, 
however, 2/2 = 00 , 

R = ^, 

2/1 

making R a finite quantity for any length from Ai. 

If the point were taken on the rod between Ai and B, 
with lower hmit, —1/1, 



R= -Km 



\2/i 2/2/ 



and, if were taken at the middle point of the rod, it is 
evident that R = 0. 

Example 2. — Attraction of a Spherical Shell at a Point. — 
Find the resultant attraction of a spherical shell of uniform 
density and small uniform thickness on a particle of unit 
mass, ikf ' being the mass of the shell. 



ATTRACTION. LAW OF GRAVITATION 



369 



(a) Let the point P outside the shell be the position of 
the particle. 

Let 7 be the density and t the thickness of the shell," its 
center, and a the radius ON; let NP = r and OP = d. If 
the circle be revolved about OP as an axis through an angle 
2 TT, a thin spherical shell of thickness t will be generated, and 
an elementary volume will be generated by the elementary 
area at N, whose mass will be AM' = yt ' 2 ira^ sin d Ad, 
approximately. 




The attraction of the elementary mass at N for the particle 
of unit mass at P is 

Kyta Ad 



A^F 



(1) 



This attraction may be resolved at P into a component X 
along PO and a component Y perpendicular to PO. To 
every elementary mass at N there is a corresponding mass 
at N', whose attraction at P is X along PO, and — Y per- 
pendicular, which neutralizes Y. Hence the attraction of 
AM' is along PO, and is given approximately by 



AF 



Kyt'27ra'^smdAe 



COS0. 



(2) 



From the geometry of the figure, 

r2 = a2 + d2- 2adcose, 



370 INTEGRAL CALCULUS 

which differentiated gives 

T dv 
» rdr = ad^inBdB', .'. sin0 



and from the figure, cos <^ = 



ad'dd' 
d — a cos 



r 
Substituting these values in (2) gives exactly, 

at /d^ — a^ + r^\ , ,„, 

dF = KjT ^ (^ ~~^—j dr; (3) 

hence, F = ii:^. ^, X_^ ( ^ )dr 






(4) 



It follows from (4) that the attraction is the same as 
though the mass of the shell were concentrated at its center. 
It follows also that a sphere, which is either homogeneous or 
consists of concentric shells of uniform density, attracts a 
particle without the sphere as if the mass of the sphere were 
concentrated at its center. This law holds almost exactly, 
for bodies slightly flattened at the poles, if the particle is not 
too close to the attracting body. Since both these con- 
ditions exist in the case of the Earth and other members of 
the Solar System, this law has important apphcations. 

(6) Let the point P' inside the shell be the position of the 
particle. The equation (3) in case (a) is true for this case 
too, but the Umits for r are now a — d and a -\- d. Hence, 

that is, the resultant of all the attractions of the elementary 
masses of the spherical shell on a particle within the shell 
is zero. 

(c) Let the point where the particle is be on the surface 
of the shell. 



ATTRACTION. LAW OF GRAVITATION 371 

In (4) making d = a gives 

F = 4:KyTrt = ^^' (6) 

Corollary. — If a particle be inside a homogeneous sphere 
at a distance d from its center, all that portion of the sphere 
at a greater distance from the center than the particle has 
no effect on the particle, while the remaining portion attracts 
the particle in the same way as if the mass of the remain- 
ing portion were concentrated at the center of the sphere. 
Thus the attraction of the sphere on the particle is 

^ = ~~d^=~~Y~' ^^^ 

that is, within a homogeneous sphere the attraction varies 

as the distance from the center. The attraction of a sphere 

of mass iif on a particle at the surface is from (7), making 

d = a, 

p 4j^ KM ,_, 

F = -^Kirya = -^- (8) 



Hence, the attraction for an external particle is 

KM 



(9) 



where d is the distance from the particle to center of sphere. 

Note. — The propositions respecting the attraction of a 
uniform spherical shell on an external or internal particle 
were given by Newton (Principia, Lib. I, Prop. 70, 71). 

It was in 1685, nineteen years after he had conceived the 
theory of universal gravitation, that he completed the veri- 
fication of the theory, by proving that a sphere in which the 
density depends only upon the distance from the center 
attracts an external particle as if the mass of the sphere were 
concentrated at its center. Thus was the great induction 
by this supplementary proposition finally estabUshed. 



372 INTEGRAL CALCULUS 

Example 3. — Attraction of the Earth. ^ — I. Find the 
relation between the attraction of the Earth on a body at 
the surface and at a point h feet above the surface. 

Taking the Earth as a sphere whose density is a function 
of the distance from the center, R as the radius, and F and F' 
as the Earth's attraction upon the body at the surface and 
at h feet above the surface, 

F/F' =(R + hy/R' (by Ex. 2, (8) and (9)), 
or F' = FRy{R + h)\ (1) 

If /i is a small fraction of R, then approximately, 

r = F{1-^ h/R)-^ = F (1 - 2 h/R). (2) 

Since the '^ weight" of a body is the force with which the 
Earth attracts it, the equations (1) and (2) give the relation 
between the weight of a body at the surface and at a height 
h feet above the surface. And, if g and g' are the values of 
the acceleration of gravity at the surface and at the point h 
feet above the surface, since F/F' = g/g', the equations give 
the relation between g and g' also. 

(a) Find approximately at what height above the surface 
will the weight of a body be tV of one per cent less than at 
the surface. 

Taking the mean radius of the Earth as 20,902,000 ft., 

F'/F =l-2h/R=l- 1/1000; 

, R 20,902,000 ,^.r,,f .• 
••• ^^2000^ 2000 -lM51feet. 

Corollary. — A mass which at the surface weighs one pound 
at 10,451 ft. will weigh 0.999 lb. 

(6) Find how much the value of g is changed by a change 
of elevation of one foot above the surface. 

¥ = 7 = 1 - IT = 1 - 2Pil;000 = 1-0.0000000957. 

* This example is based on examples Lq Hoskins's Theoretical Me' 
chanics. 



ATTRACTION. LAW OF GRAVITATION 373 

The value of g for different latitudes and elevations is given 
by the following formula, in which g is in feet per second, I 
is the latitude, and h the elevation in feet above sea level : 

g = 32.0894 (1 + 0.005243 sin^ I) (1 - 0.0000000957 h). 
This gives 

g = 32.0894 at the equator at sea level, and 

g = 32.174 at 45° latitude at sea level ; 

this latter value, g = 32.174 ft. per sec. per sec. is the 
standard value. 

II. Find the relation between the attraction of the Earth 
on a body at the surface and at a point h below the surface, 
(a) Taking the Earth as a sphere of uniform density of radius 

R, 

Y"IF = {R- h)/R =l-h/R (by Cor. Ex. 2), (3) 

where F and F" denote the attraction at the surface and at 
h below the surface. 

Corollary. — Under these conditions, the weight of a body 
and the value of g would decrease with the depth h below the 
surface. 

(6) Taking the Earth as a sphere whose density is a 
function of the distance from the center, let y denote the 
mean density of the whole Earth and 70 the mean density 
of the outer shell of thickness h. 

Let M be the mass of the whole Earth, M'' that of the 
inner sphere of radius R — h, m the mass of the attracted 
body, F and F'' the attraction at the surface and at h below 
the surface. Then F is equal to the attraction between two 
particles of masses M and m whose distance apart is R, and 
F" is equal to the attraction between two particles of masses 
M'' and m whose distance apart is R — h. That is, 

F = KMm/R\ F'^ = KM"m/{R - h)'; 
, F'' M" I R V 



374 INTEGRAL CALCULUS 

Now M = ^TvR'y; M - M" = Ittto [R' ~ {R - hf]; 



M 
M 



--'-?['-(V)>(-?)-?(v;. <« 



which substituted in equation (1), gives 

^ = ('-?)(s^jH-(V> '« 

If /i is a small fraction of R, equation (6) may be reduced 
to the approximate formula, 

Corollary. — If the mean density of the outer layer of the 
Earth is less than two-thirds the mean density of the whole 
Earth, the weight of a body increases as it is taken below 
the surface of the Earth. (See Ex. 1, Art. 171.) 

The mean density of the Earth being taken as 5.52 and 
that of the layer near the surface as 2.76, about the density 
of the rocks, makes 70/7 = i and equation (7), 

F'^F = W'lW = g"lg = 1 + V2 R. (8) 

That the weight of a body increases as it is taken below the 
surface has been shown by actual trial. From (8), the depth 
to which a body must be taken in order that it gain j^q of 
one per cent in weight is approximately, 

, 2X20,902,000 .,„„,. 

^- 10,000 ^^^^Q^^- 

that is, a mass weighing a pound at the surface will weigh 
1.0001 lb. at a depth of 4180 ft. below the surface. 

Compared with case I, it may be seen that, under the 
conditions, for the same value of /i, the gain in weight is one- 
fourth as much as the loss in weight when the body is above 
the surface, the same ratio of change applying to the value 
of gf also. 



VALUE OF THE CONSTANT OF GRAVITATION 375 

191. Value of the Constant of Gravitation.* — From the 
foregoing as to the attraction of a sphere, it follows that the 
formula for the attraction of two particles, 

F = k'^ [(1) Art. 190] 

will apply to two spheres, which are either homogeneous 
throughout or composed of a series of concentric shells, each 
one of which is of uniform density, m and m' being the masses 
of the spheres and r the distance between their centers. 

By measuring the force of attraction between two spheres 
of known mass and distance apart, the value of K the constant 
of gravitation has been found. As stated in Art. 190, its 
numerical value will depend on the units used for the other 
quantities in the equation. The relation between the 
constant K and the mass of the Earth, taking the Earth as a 
sphere whose density is a function of the distance from the 
center may be shown as follows. 

Let the units be the British gravitation units, and let R 
be the radius of the Earth in feet, M its mass, 7 its mean 
density. Consider the attraction of the Earth on a body of 
mass m at the surface. By the formula (1) of Art. 190, the 
value of the attraction is KMm/R^; but (since the unit force 
is the weight of a pound mass) expressed in pounds force, 
its measure is m. Hence, m = KMm/R'^ or 

KM = R\ (1) 

Since the value of R is known, either K or M can be found 
when the other is known. Putting for M its value in terms 
of 7, 

K'^irR^y = R' or Ky = ^' (2) 

\ Arts. 191 and 192 are based on Articles in Hoskins's Theoretical 
Mechanics. 



376 INTEGRAL CALCULUS 

Taking y = 345 lbs. per cu. ft. and R = 20,900,000 ft. gives 

K = -r^ = 3/(47r X 20,900,000 X 345) = 3.31 X lO"". 
4 7rii7 

Otherwise, if the value of K, found by direct measurement 
of the attraction of two spherical bodies, is substituted in (2) 
the value of 7, the mean density of the Earth is found to 
be 5.527. The density of water being unity, and its weight 
62.4 lbs. per cu. ft., the mean density of the Earth is about 
345 lbs. per cu. ft., as used above. 

192. Value of the Gravitation Unit of Mass.* — As 
stated in Art. 190, the force with which two particles of unit 
mass at a unit distance apart attract each other is equal 
to X, the constant of gravitation; this is evident from the 
equation, 

F = k'^- [(1) Art. 190.] 

Let m pounds be the mass of each of two particles which, 
when one foot apart, attract each other with one pound force. 
Substituting K = 3.31 X 10~^\ as given above, putting 
F = 1, m = m\ and r = 1; gives 

m = 1/Vk = 173,800 lb. 

If a mass equal to 173,800 pounds be taken as the unit 
mass, the constant K becomes unity and the formula for 
attraction is then 

p, ^ mm' 

The gravitation unit of mass is thus shown to be a mass equal 
to about 173,800 pounds, distance being in feet and force in 
pounds-force. 

193. Vertical Motion under the Attraction of the Earth. 
— Let the Earth be taken as in Example 3, Art. 190, r as 
the radius and s the distance of the moving particle from 

* See Footnote on page 375, 



VERTICAL MOTION 



377 



the center 0. Taking distance, velocity, and acceleration 
as positive outward, then, as in (1) Ex. 3, 



F g s' 



or g' = '-^ 



Since -rr = « and -j: = v, ehminating dt 
at at 




gives V dv = a ds. 

Hence, a =■ —g' = —^, neglecting air re- 

o 

sistance, which gives 

I vdv = j — grh~^ds; 

integrating, ^ = S^^'«"M^ = S'^' 1^^: " - ) ; 

then, v^ = 2gr^(^^-fj (1) 

gives the velocity of a particle towards the Earth from any 
distance s. 

For the velocity acquired by a body in falhng to the surface 
from a height h, put s = r + /i in (1), giving, 



2gr' 



\r r -\-hl 



= 2 gh, approximately. 



(1') 

(2) 



if h is small, which is the formula when g is constant, as it is 
taken near the surface. When - is small, putting 

rrj-o+r— (^e)"-©"+---. 

(2) becomes 

v'=^2gh{l- (h/r) + (h/ry - (h/rY +•.•]. (3) 



378 INTEGRAL CALCULUS 

it- 

By taking any number of terms of this series, an approxi- 
mate result may be gotten as nearly correct as desired. If, 
in (1), s = 00 , z; = V2 gr; so if a body fell toward the Earth 
from an infinite distance, its velocity, neglecting air resist- 
ance, would be V2 gr = 6.95 miles per second, for r = 3960 
miles. If falling from a finite distance s, the velocity must 
be less than this. Hence, a body can never reach the earth 
with this velocity; and if air resistance is considered, the 
velocity for s = oo is less than a/2 gr. If projected outward 
with velocity a/2 gr and air resistance be neglected, the body 
would go an infinite distance. This velocity is called the 
critical velocity or velocity of escape, for under the conditions 
it is supposed that certain particles of the atmosphere may 
escape from the attraction of the Earth. 

In this connection, it is to be recalled that due to the 
Earth's rotation, there is at its surface a centrifugal force 
mgr/289, exerted by a particle of mass m, which lessens the 
value that g would otherwise have. 

194.* Necessary Limit to the Height of the Atmosphere. 
— The centrifugal force of a particle of mass m on the surface 

of the Earth is mojV = -pr^, and at a distance s from the 

center it would be mco^s = ^Ji . The Earth's attraction at 

289 r 

TYior'^ 
that distance being — ^, in order that the particle be re- 
s 

tained in its path these two forces must equal each other; 
mgs _ mgr^ 
•*• 289r ~ ~7~' 
or s^ = 289 r^, 

hence s = a^289 r = 6.6 r 

= 26,000 miles approximately; 
that is, a height above the surface of about 22,000 miles. The 
actual height of the atmosphere is probably much less than 
* Bowser's Hydromechanics. 



MOTION IN RESISTING MEDIUM 379 

this. The estimates of the height by various scientists have 
been very divergent — from 40 miles to 216 miles; but the 
latter appears to be the most hkely, for meteors have been 
observed at an altitude of more than 200 miles and, as they 
become luminous only when they are heated by contact with 
the air, this is evidence that some atmosphere exists at that 
height. It is supposed that at a height much less than 5.6 r, 
the air may be hquefied by extreme cold. 

195.* Motion in Resisting Medium. — Consider the 
motion of a body near the surface of the Earth under the 
action of gravity taken as a constant force and the air taken 
as a resisting medium of uniform density, the resistance 
varying as the square of the velocity. 

Let a particle be supposed to descend towards the Earth 

from rest, and let s be the distance of the particle from the 

starting point at any time t, gk'^ the resistance of the air on 

a particle for a unit of velocity — gk^ being the coefficient 

of resistance. The resistance of the air at the distance s from 

/dsV 
the origin will be gk'^i-r-j , acting upwards, while g acts 

downwards, the mass being a unit. 
The equation of motion is 

/^o\2 

(1) 



r\T 


dh ..fdsV 


or 

Integrating, 


1 , dt 
^ "dt 



t = 0,v = 0, giving C = 0. 

* Bowser's Analytic Mechanics.' 



380 INTEGRAL CALCULUS 

Passing to exponentials, 

ds _1 e^^* — e~^°* 
dt~ k e^ot _|_ Q-kgt ' 



(2) 



which gives the velocity in terms of the time. To get it in 
terms of the space, from (1), 



2gVds; 



.-. log [l - 1<? gj] = -2gh\ s = 0, z) = 0; C, = 0, (3) 

which gives the velocity in terms of the distance. Also, 
integrating (2); 

gkh = log (e*^' + e-*^0 - log 2; 

which gives the relation between the distance and the time 
of falhng through it. 

As the time increases the term e~^^' diminishes and from 
(5) the space increases, becoming infinite when the time is 
infinite; but from (2) as the time increases the velocity 
becomes more nearly uniform, and when ^ = oo , the velocity 
= 1/k) and although this state is never reached, yet it is 
that to which the motion approaches. 

196. Motion of a Projectile. — If a body be projected 
with a given velocity in a direction not vertical and be acted 
on by gravity only, neglecting the resistance of the air, it is 
called a projectile. The path, called the trajectory, will result 
from a combination of the motions due to the velocity of 
projection and to g, the vertical acceleration of gravity. 
Let the plane in which a particle is projected with a velocity 



MOTION OF A PROJECTILE 



381 



V be the plane of XY, and let the hne of projection be inclined 
at an angle a to the x-axis, making v cos a and v sin a the 
resolved parts of the velocity of projection along the axes of 
X and y. 

Y 



s 


D 


n' 






/ 








/ 




4 




>^2 


p 






\ 


/ ^ 


i 


c 




B ^ 



Let {x, y) be the position of the particle P at the time t) 
then, since the horizontal acceleration is zero and the vertical 
acceleration negative, 

d}x _ d'^y _ 

d^~ ' d? ~ ~^' 
Taking the first and second integrals of these equations, 
determining the value of the constants of integration corre- 
sponding to ^ = and t = t, gives 



dx 

-fj = V cos a ; 

dt 



dy 

-^ = vsma — gt; 



V cos at; y = V sin at — \ gf^. 



(1) 

(2) 



Equations (1) and (2) give the coordinates of the particle 
and its velocity parallel to either axis at any time t. 
Eliminating t between equations (2), gives 

,2 



y = X tan a — 



gx' 



(3) 



2 v^ cos^ a ' 

which is the equation of the trajectory, and shows that the 
path of the particle is a parabola. 
Putting equation (3) in the form 



2 v^ sin a cos a 
9 



x = 



2 v^ cos^ a 



y> 



382 INTEGRAL CALCULUS 

or 

/ v^ sin a cos a V 2v^cos^a/ v^sm^a\ 

(' -g — ) = — ~[y-^r)' ^'^ 

and comparing this with the equation of a parabola, 

{x-hy== -2p{y-k), 

it is seen that: 

V sin ci cos (Y 
the abscissa of the vertex = ; (5) 

v^ sm^ ex 

the ordinate of the vertex = — ; (6) 

2g 

the latus rectum = (7) 

g 

By transferring the origin to the vertex, (4) becomes 

X^=-?^^^y, (8) 

which is the equation of a parabola with its axis vertical and 
the vertex the highest point of the curve. 

The distance between the point of projection and the point 
where the projectile strikes the horizontal plane, called the 
Range, is 

OB = X = , (9) 

9 

when y = 0, from (3); which is evident geometrically, since 

OB = 2 OC; that is, the range is equal to^wice the abscissa 

of the vertex. 

It follows from (9) that the range is greatest for a given 

velocity of projection, when a = 45°, in which «iase the 

range = — . It appears from (9) that the range is the same 

if 

for the complement of a as for a. The greatest height CA 
is given by (6) which, when a = 45°, becomes v'^/4: g. 
The height of the directrix, 

^7^ ^4 . A T^ v^ silica , 12v^cos^a v^ 

CD = CA-^AD = — TT h T = :^- 

2g 4: g 2g 



MOTION OF PROJECTILE IN RESISTING MEDIUM 383 

Hence, when a = 45° the focus of the parabola is in the 
horizontal Hne through the point of projection, for then 
CA = i CD. 
To find the velocity V at any point of the path, from (1), 

>-(D'+(S)" 

= v^ cos^ a + {v'^ sin^ a — 2 1; sin agt + gH"^) 

= v''-2gy, or ^ = ^ - y = MS - MP = PS. 
2g 2g 

72 
Since ^^ is the height through which a particle must fall 

from rest to acquire a velocity V, it follows that the velocity 

at any point P on the curve is that which the particle 

falHng freely through the vertical height SP would acquire; 

that is, in falUng from the directrix to the curve; and the 

velocity of projection at is that which the particle would 

acquire in falHng freely through the height CD. 

For the time of flight, put ^ = in (3) and solve for 

2 v^ sin a cos a ^ - ■, j- • ^ t ^ • ^- <. 

X = J which divided by v cos a gives, time of 

2 V sm oi 
flight — ; or in (2) put y = and solve for t, giving 

r^ J ^ 2?;sinQ: , » 
t = and t = , as before. 

g 

197. Motion of Projectile in Resisting Medium. — If 

the resistance of the air is taken to vary as the square of the 
velocity and the angle of projection is very small, the pro- 
jectile rising but a very Uttle above the horizontal, the 
equation of the trajectory above the horizontal line can be 
found. Thus the equation 

gx^ gkx^ 

y = X tan a — p—^ — ^ 7^-^ ^ * • * > 

^ 2v^ cos^ a Sv^ cos^ a ' 

may be derived under such conditions; where the first two 



384 INTEGRAL CALCULUS 

terms represent the trajectory neglecting air resistance, as 
found in (3), Art. 196. 

For the high velocities of cannon-balls the trajectory is 
found to be very different from the parabolic path and the 
range much less than that deduced for it. 

Experiments show that the angle of projection for greatest 
range is about 34°, rather than 45°, as deduced for the para- 
bohc path. 

The simplest formula for making out a range table is 
H6he's: 

qx^ / 1 , kx\ 

2cos2a\i;o Vo/ 

where k = 0.0000000458 -, d being the diameter of the 

w 

projectile in inches, and w its weight in pounds. 

In addition to the resistance of the air, allowance has to 
be made in firing for the drift, that is, the tendency for most 
projectiles to bear to the right upon leaving the gun, due to 
the right-handed rotation given to the projectile. 



CHAPTER VIII. 
INFINITE SERIES. INTEGRATION BY SERIES. 

198. Infinite Series. — When a series consists of a 
succession of terms whose values are fixed by some law and 
the number of its terms is unlimited, it is an infinite series. 

Let Ui, Wy Us, . , . be an infinite succession of such values 
and let the sum of the first n of these values be denoted by 
Sn, that is, 

Sn = Ui + Ih -\r '•'-{- Un- (1) 

When n becomes infinite, then the infinite series is 

U1 + U2 + U3+ ' ' ' , (2) 

If Sn has a definite limit as n becomes infinite, that limit 
is called the value of the infinite series, and the series is said 
to be convergent. 

If Sn has no definite limit, either oscillating between two 

finite values or increasing in value beyond any finite value, 

the series is said to be divergent. Thus the geometrical series, 

a -\- ar + ar"^ -{- ar^ -{- • • • (3) 

is convergent only when r is between — 1 and +1, for 

Sn = a-\-ar + ar^-\- • • • -\-ar''-^=-\ - = - :; , 

1 — r 1 — r 1 — r 

and 

lim Sn = lim —^ = ^ , when — 1 < r < 1. 

n=oo n=oo i- r i r 

Since *S„=oo = — ^^ = 00 , when — 1 > r = 1, 

1 — r 

and oscillates between and a when r = — 1, the series 
for those values of r is divergent, *Sn=oo having no definite 
limit. 

385 



386 INTEGRAL CALCULUS 

The terms of a series may be functions of some variable x; 
then the series is said to converge for any particular value 
of this variable, say x = a, when, if x is replaced by a in each 
term, Hm Sn (a) exists. When the corresponding limits exist 

n=oo 

for all values of x in a certain interval, say from x = a — h 
to X = a + A, the series is said to converge throughout the 
interval and to define a function in the interval. 
Example 1. — Let the given series be 

1+ X + ^2+ a;3 + . . . + x^-^ + • • • . • (1) 

Here lim Sn (x) — hm — — — = -— — -, for — 1 < x < 1 ; 

n = oo n=oo ji X L X 

and within the interval x = — ltox= +1, not including 

the end values, the series defines the function f{x) = 

1 — X 

For the end values x = — 1 and x = + 1, and for all values 
beyond, the series is divergent and does not define a function. 
Example 2. — Similarly the given series, 

l-x + x^-x'-i- • • • + i-iyx^-'^ + • • • (2) 

has hm Sn{x) = , for — 1 < x < 1, 

n= oo J- I X 

and within the interval x = —1 to x = +1, the series de- 
fines the function /(x) = — -- — 

JL "Y" X 

For the end values x = —1 and x = -\-l, and for all 
values beyond, the series is divergent and does not define a 
function. 

199. Power Series. — An infinite series of the form 

ao + aix + CL2x'^ + asx^ + • • * + anX"" + • • • , (1) 
where ao, ai . . . , an, etc., are constants, is called a power 
series in x. One of the form 

ao -\- ai{x — a) + a2 {x — ay + as (x — a)^ + • • • 
+ a. (x - a)- + • • • (2) 

is called a power series in {x — a). 



POWER SERIES 387 

The series (1) and (2) of the preceding examples (Art. 198), 
are power series in x, representing the functions and 

-L jC 

for certain values of x. Such series are of importance 



l+x 

because of the frequency with which they occur and the 

special properties which they possess. 

For instance, the sum of a few terms of an infinite series 
representing some function may be a very close approxima- 
tion to the value of the function. Thus, if in the series (1), 
(Ex. 1, Art. 198), x = \, the well-known series converging 
to the value 2 results : 

^ =2=Hh~ + i + i+ • • • +^+ • . . . (3) 



l-x ■ 2 ■ 4 ' 8 

If the terms In (1) (Ex. 1, Art. 198) after x^-i be neg 
lected, the error would be, 

k 

x^ + a;^+5 + • • • + x^ + ^ 



1 -x 



X 



This error, , would be very small compared with the 

x X 

value of the function, :; , and would decrease as k was 

I — X 

increased; that is, a closer and closer approximation would 
be made to the value of the function the greater the number 
of the terms retained. For the particular value x = \, it 
may be noticed that the error made in stopping with any 
term is exactly the value of that term. 

For smaller values of x a very much closer approximation 
would be made even when only a few of the terms are taken. 
This method of approximation is practically useful when the 
exact value of the function is unknown or does not admit of 
exact numerical expression, for examples, the numbers e and 
TT; the logarithms of numbers, and the trigonometric func- 
tions of angles in general. In Articles to follow power series 
for such functions will be given. 



388 INTEGRAL CALCULUS 

200. Absolutely Convergent Series. — A series the abso- 
lute values of whose terms form a convergent series is said to 
be absolutely convergent ; other convergent series are said to 
be conditionally convergent. For example, the series 

1 - 4 + i - i + • • • (Ex. 2, Art. 198), (1) 

is an absolutely convergent series, since it converges when 
all terms are given the positive sign, as (3), Art. 199. On 
the other hand, the series 

1 - i + 4 - i + J . . . (2) 

is conditionally convergent, since the series 

resulting from making all terms positive, is divergent. 
The series (3) may be seen to be divergent in the form, 

^ Vn + ;r+I + • • • + 2{n-l) + • • • ' 

for the sum of the terms in each of the parentheses is greater 
than J, and as the number m of such groups that can be 
formed in the given series is unlimited, 

>Sn=oo > (m X i) = 00. 
This result reveals the important fact that while the defini- 
tion of an infinite convergent series, requiring Sn=oo to have 
a definite limit (Art. 198), makes lim t^n = a necessary 

condition, that condition is not sufficient to insure conver- 
gency. In other words, a series is not necessarily convergent 
when the terms themselves decrease and approach zero, as 
the number of terms increases without limit. For the series 
(3), the condition is fulfilled; the nth term approaches zero 
as a limit, as n increases without limit, and yet the sum of the 
first n terms has no limit and the series is divergent. 



ABSOLUTELY CONVERGENT SERIES 389 

When, however, the terms of the decreasing series are 
alternately positive and negative the condition is sufficient 
to prove convergency. Thus the series (2) being such a 
series is convergent, though not absolutely so. This series 
may be put in the forms, 

(1 - 4) + (4 - }) + a - i) + • • • , 

or 4 + T 2 + ^V + • • • , 

1 - (4 - 4) - (i - «... , 

or 1 - 4 - 1^0 - • • • , 

where it can be seen that the sum of n terms of the series is 
greater than | and less than 1. It will be shown further on 
that the limit of the sum is log 2 = ;0.69 . . . (Ex. 1, Art. 203). 
While it is thus seen that the series (2) is conditionally 
convergent, the following series is absolutely convergent : 

1 - i -4- 1 - i -J- 1 - . . (A\ 

2^ 3^ 4^ 5^ ' ^^ 

since 1 + 2~2 + 3~2 + 4! + 5^ + * * ' (^) 

is convergent, as may be shown by comparison with the series 

1+1+1 + 1+1+-... 

2 2^ ^ 2^ ^ 2^ ' 

or 1 + 4 + 4 + 4+---, (3), Art. 199, 

known to be convergent. 

The series (5) is the more general series 

lp'2^3p4p^*np' ^^ 

when p = 2; and the series (3) of this Article, called the 
harmonic series, is the series (6) when p = 1. 

This series (6) is, therefore, divergent for p = 1; for 
p < 1, every term after the first is greater than the corre- 
sponding term of the series (3), hence (6) is divergent in this 
case also. 



390 


INTEGRAL CALCULUS 


For p 
1 + /1 


> 1, compare 


5P -r gp -r y^y 




with 
P ^ V2^ 


+ 2^) + (r.+ 


/I 
i + l+n 


1 \ 

+ 15.) + 






Hk^- 


■ +^.) + 



(6) 



• (7) 

There is the same number of terms in the corresponding 
groups of the two series and the sum of the terms in those of 

(6) is in each group less than the sum in those of (7). Now 

(7) may be put in the form 

Ip^ 2p 4:P S^ ' 

2 

a geometrical series, whose ratio, — , is less than unity. Hence 

by (3) of Art. 198, (7) is convergent and consequently (6) 
is convergent. 

The series (6) and the geometrical series are useful as 
standard series, with which others may often be compared 
to test convergency. 

201. Tests for Convergency. — It has been seen that a 
conditional test is that in every convergent series the nth term 
must approach zero as a limit, as n is increased without limit. 

This condition is involved in and may be deduced from 
the definition (Art. 198), that the infinite series 

ui-\-u2 + m-\- ' ' • (1) 

is convergent, if lim Sn exists. 

n=oo 

For since 

Sn = Sn-1 + Un, lim Sn = Hm Sn-1 + li^l Un = Hm Sn-1, 

n = oo n=oo n=oo n=oo 

if lim Sn exists ; .*. lim Un = 0. The converse is not true. 

n=oo n=<x> 



TESTS FOR CONVERGENCY 391 

In the same way it may be shown that the Remainder after 
n terms of a convergent series must approach zero as a hmit 
as n becomes infinite; thus, letting Rn denote the series of 
terms after the nth, 

Rn = Uu+i + Un+2 + Un+3 + * • ' . 

Now if S denotes the value of the series (1), it is the limit of 
the sum of the terms in the series; that is, 

S = ]mi{Sn + Rn) = limAS^„; 
:. lim Rn = 0. 

n=oo 

For example, in (3), Art. 199, Rn = (i)""^; .*. Hm Rn = 0. 

n=oo 

It was seen in (3), Art. 200, that although the nth term 
approaches zero, the series is not convergent, hence the 
remainder after n terms does not approach zero as a limit in 
that series, as it is a divergent series. 

For convergency, lim Rn = is & decisive test; lim Un = 

n=oo n=oo 

is a decisive test only when the terms alternately have dif- 
ferent signs; it is a conditional test when all the terms have 
the same sign. For divergency, lim Un not equal to zero is a 

decisive test. Hence, when lim Un = 0, unless the terms 

n = oo 

alternately have different signs, the test is indecisive. 

The Comparison Test. — It may often be determined 
whether a given series of positive terms is convergent or 
divergent, by comparing its terms with those of another 
series known to be convergent or divergent. The method 
of applying this test and the standard series useful for com- 
parison have been given in the preceding Article 200. This 
test is often available when other tests fail to be decisive. 

The Ratio Test. — A given series 

U1+U2+UZ+ ' ■ • + Un^l +Un+ ' ' ' 

is convergent or divergent, according as lim — — is less or greater 

n=(X) Un—1 

than 1. 



392 INTEGRAL CALCULUS 

This test applies when some of the terms of the series are 

negative as well as when they are all positive. It is no test, 

u 
however, when Um — — = 1 ; in that case other tests must 

n=ao Uji—l 

be appHed. 

For example, let the given series be 



x^ , oc^ 



>n-^l 



Here lim — - = lim ( — , / 7 777 ) = Hm - = 

n=ooWn-i n=oo\n!/ (n — 1) !/ n=oo n 

for any finite value of x. Hence the series (2) converges for 
all finite values of x. 

This series (2), as given in Art. 36, is the expansion of 6^; 
and, when x = 1, the limit of the series is the number e, in 
Art. 34 (see Ex. 5, Art. 215, also). The ratio test is found 
to be true by comparing the given series with the geometrical 

series; hence, when lim — — = 1 and the test fails to be de- 

n=<x>Un—l 

cisive, recourse is to the comparison test, as shown in Art. 
200, for the harmonic series 

EXERCISE XLII. 

1. Find whether the following series are convergent or divergent: 

(1) 1 + i + i + ^ + • • • . 

(2) I + i + 1 + I + • • • . 

^^^ ]T2 + 2^3 + 3^+ 475 + • * * • 
V^; 1 -r 2 ^ 22 ^ 23 ^ 2^ ^ 
(5) 2 + 3'! + 4! + 5'!+ * ' *• 

02 Q3 44 

(«) l + 2! + 3! + 4!+---- 



CONVERGENCE OF POWER SERIES 



393 



(7) I - I + f - t + 



TT 



+ 



(9) 



1 



n2 + l 
1 



= + 



1 + VI 1 + V2 1 + V3 

(10) 1 + ^ + 1 + 1^ + ^+ .... 

2. Examine the following series for convergency: 
(1) logf-logf + logt-logf + . . • . 



(2) sec ;r — sec -7 + sec p 
o 4 5 



secg + 



(3) sin2| + sin2| + sin2^ + sin2|+ .... 

202. Convergence of Power Series. — A power series in 
X may converge for all values of x, as in (2), example of the 
Ratio Test; but generally it will converge for certain values 
of X and diverge for others, as in Examples 1 and 2, Art. 198. 

Applying the ratio test to the power series, 

ao + aix + aix^ + asx^ + • • • + anX" + • • • . (1) (Art. 199.) 



= ; lim 

'^n— 1 Cln—l n=oo 



Un 



= lim 



anX 

Oin-l 



X 1 lim 



ttn-l 



The series (1) is convergent or divergent according as 



I X I lim 



OLn 



ttn-l 

that is, according as 

\x\ < Mm 

n=oo 

The case 1x1 = lim 



<1, 



an-i 



an 

Cin-l 



or 



or 



X I lim 



a„ 



Cln—l 



>i; 



X I > lim 



(In-l 



an 



is undecided by this test. 



When, however, a power series is convergent for any value 
of X, say a, it is absolutely convergent for all values of x such 
that I X I < I a; |. 

For example, given the series 
l + 2x + 3x2 + 4x3+ . . . nx"-i + (n + 1) X" H ; (2) 



394 INTEGRAL CALCULUS 

here = — r-r , lim — -— = iim r = 1 ; 

an n-\-l n=oo n-\-l n=oo -, , 1 

n 
hence (2) is convergent or divergent, according as 

|a;| < 1 or |a;| > 1; 

that is, (2) is convergent when — 1 < x < 1, and the interval 
from —1 to +1 is the interval of convergence. Here the 
interval does not include the end values, (2) being divergent 
when I oj I = 1. 

203. Integration and Differentiation of Series. — A 
power series has the important property that, when the 
variable of the function is restricted to the interval of con- 
vergence, it is possible to get the integral or the derivative of 
the function by integrating or differentiating term by term 
the series which defines the function. Hence, if / (x) is 
defined by the power series, 

f{x) = ao + aix + a2X^ + • • • + anX"" + • • • ; (1) 
then 

Jf{x)dx= j aodx-{- I aixdx-\- • • ♦ + / anX'^dx-^ • • • , 

and 

dfjx) ^ d(ao) djaix) ^ ^ ^ djarX") . 
dx dx dx dx ' 

when the restriction necessary to insure convergence is placed 
upon the value of x. 

Example 1. — For — 1 < x < 1, 

^_L_ = 1 _ X + x2 - x3 + . . . . (Ex. 2, Art. 198.) (2) 
Hence, 

JT— j — = / dx— I xdx-\- I x'^dx— I x^dx-jr ' ' ' , 
1 + a; Jo Jo Jo Jo 

thatis, log(l+a:) = a:-|' + f-|'+ • '• . . (1) 



INTEGRATION AND DIFFERENTIATION OF SERIES 395 

This is the logarithmic series with base e, and is true for 
— 1 < X = 1; that is, it is true for values of x within the 
original interval of convergence, including the end value 1; 
but for the other end value — 1, it decreases without limit. 
: On putting x = 1 in (1), 

log 2=l-§ + J-i+---= 0.69 .... (See Art. 200.) 

On putting a; = — 1 in (1), 
logO= - (l+i + i + i+ • • • ) = -00. (See Art. 200.) 

In the same way, for — 1 < a; < 1, the integration of 

-J_ =i-^x + x^-\-x^-\- ' • • (Ex. 1, Art. 198) (1) 

gives \og(l — x) = —X — ^x^ — ix^ — Ix"^ — • • • , (2) 

which may be gotten by putting —x for x in (1). 

This logarithmic series is true for —1 ^ a: < 1; that is, 
it is true for values of x within the interval of convergence, 
including the end value —1; but for the other end value 1, 
it decreases without hmit. 

For X = — 1, (2) gives log 2 as above, and for x = 1, 
log as above. Hence by neither series can the logarithm of 
a number greater than 2 be found. By a combination of 
the two series the logarithm of any number can be found. 

By subtracting (2) from (1), 

1 -\- X 
log (1 + x) - log {1- x) = log — — : 

i X 

= 2{x + ^ + f+- ■ •), for|a;|<l. (3) 

For X = ^, 
lo,]±l^lo^2=2{l+^,+^,+^,+ - ■ •)=0.6931 .... 

For X = i, 
logi±| = log3 = 2(i+3i23+5^+7^+- • ■)= 1.0986. . .. 



396 INTEGRAL CALCULUS 

This series (3) converges very much more rapidly for 

values of x less than 1 than the series (1), which converges 

so slowly that 100 terms give only the first two decimals 

correctly for the log 2, while (3) gives four decimals correctly 

taking only four terms of the series. Any number may be 

1 + a; 
put in the form -— — , but it is necessary to calculate directly 

JL X 

the logarithms of the prime numbers 2, 3, 5, 7 only, as the 
others can be expressed in terms of these. Thus, 

1 + - 5 5 

log Yzri = ^^S 3 , and then log 5 = log - + log 3; 

and again, 

1 + - 7 7 

log :j 1 = log -, and then log 7 = log ~ + log 5. 

To get the common logarithms whose base is 10, multiply 
these natural logarithms by 0.4343 . . . , the modulus of 
the common system. (See Art. 38 and Ex. 6, Art. 215.) 

Example 2. — For — 1 < a; < 1, 

-A_ = 1 + ^ _|_ ^2 _|_ ^3 _|_ . . . , (Ex. 1, Art. 198.) (1) 

J. X 

By differentiation, 

^ =l+2a;+3x^+4xH • (See (2), Art. 202.) (2) 

(1 X) 

By differentiation again, 

^j4^=^(l-2 + 2-3a: + 3.4x2+. • • ). (3) 

Hence, the general series, 

1 /i \ «, -I . , m(m + 1) „ 
(1 — x)-"^ = l+mx-\ ^^-^ — -x^ 



(1 - x)^ ' ' ' ' 2! 

I ^ (m + 1) (m + 2) ^, ^ ^^^ 



INTEGRATION AND DIFFERENTIATION OF SERIES 397 

Example 3. — For — 1 < a; < 1, by (4) of Ex. 2, or by 
division, 

Hence, l :; — ; — ^= I dx — I x^dx-{- I x^dx— • - - . 
Jq l+x^ Jq Jo Jo ' \ 

that is, 

arc tan x = x— tt + t-"***' (2) 

o o 

This is Gregory's series, named after its discoverer, James 
Gregory. 

Although series (1) oscillates when x = 1, series (2) is 
convergent and defines arc tan a; even when x = 1. On 
putting X = 1, 

, ^ TV ^ 1,11, 

arctan 1 = J = 1 - - + - - y + • • • ; 

... „.(,-i,i-.,...). 

While the value of tt may be found approximately from this 
series, the series converges so slowly that it is better to use 
other more rapidly convergent series, such as. 



and 



TT 1 1 

- = 4 arc tan - — arc tan ^^ (Machines Series), 



J = arc tan - + arc tan - • (Euler's Series.) 

4 ^ o 



Example 4. — For - 1 < x < 1, by (4) of Ex. 2, 

1 =(l-a;r^=l+^^^+^x^+^^« + 



Vl-x^ 2 '2-4 '2-4-6 

hence, 



X 



dx . ,1 x\l-3 x\l'S-5 x\ 

Vl-a;^ 2 3 2-4 5 2-4-6 7 



This series, due to Newton and used by him to compute the 
value of TT approximately, converges rapidly for a; < 1. 



398 INTEGRAL CALCULUS 

When X = i, this series gives 

. 1 TT 1 , 1 , 1-3 . 1.3.5 
arc sin ^ = - = 77+ ^ „ ^„ + 



2 6 2'2.3.23'2.4.5.25'2.4.6.7-27' 

To ten places, tt = 3.1415926536 .... 

By means of series the value of w has been carried to 700 
decimal places. 

(a^ — eV) 2 This can- 

V a^ — x^ 

not be integrated directly, but on expanding (a^ — e^x^)^ by 
the binomial theorem the terms of the resulting convergent 
series can be integrated separately. Thus, 

(a2 - e2^2)| == a-~--~~-^- ' ' ' , where e < 1, (1) 



(a2 - e^x'')^ ^^ 



r 

Jo Va^ — x^ (See Ex. 1, Exercise XXV.) 

J'^a dx _ e^ r« x'^dx __e^ f" x^ dx 
Va'^-x^ 2 a Jq Va^-x^ 8aVo Va^-x^ '^ 

= ![^/i _ !! _ /Li^Y!' __ /Lli^Y!! _ \ r2^ 

2 V 22 \2.4/ 3 V2-4.6y 5 ' ' •/ ^ ^ 

When X = asind, 

IT 

r {a' - eV)^ , ^^ = a C (^ - ^^ sin^ 0)^ ci6>; 

Jo V a^ - x^ Jo 

a Pci - e''smH)^dd 

(-©■•-(Hn-G^jf--)- «' 



7ra 
~2 



INTEGRATION AND DIFFERENTIATION OF SERIES 399 
Example 6. — Given 

r , ^"^"^"^ (See Art. 236.) 

Jo V2gih-x)i2ax-x^) 

This does not admit of direct integration, but on expanding 
it into a power series in x/2a the integral can be evaluated 
approximately. Thus, 

Jo V2g(h-x){2ax-x^) \gJoVhx-xA 2a/ ' 
by (4) of Ex. 2, 

\ gJo l^^2\2aj^2'^[2a) ^ JVa^^=^' 

by integrating, 

When h is small in comparison with a, all terms containing 
jz~ may be neglected, and the approximate value is tt t /^. 

z a \ g 

If the given integral is put in the form 

2 V - f (1 ~ ^' sin2 0)-2 dcl>; [Art. 2365 
^ Q Jo 



gJo 

then, 
2V- f\l-k^sm''(l>)-^ 

▼ <7t/o 



[by (4) of Ex. 2.] 
= 2y/^J^'(l+i/c2sin20 + ^-|/c^sin^^+ . • . ) c^0, 
By integrating, 

-'/;h(i)''-+(Hr'-+(S^)''-+ ]■ <» 



400 INTEGRAL CALCULUS 

When k is small, the approximate value of the integral is 



v~ 



agam 

' 9 

Note. — The integral forms in Examples 5 and 6 are called 
elliptic integrals. 

TT IT 

The forms / . and | Vl — e^ sin^ dcf) 

Jo Vl - Fsin2 Jo 

are known respectively as ''elHptic integrals of the first and 
the second kind." 

In the first kind k = sin ^ ; and, in Ex. 6, a is the angle 

each side of the vertical through which a pendulum of length 

a vibrates, the approximate time of a vibration being tt y -, 

as found. (See Art. 236.) Tables give values for varying 
values of a. 

In Ex. 5, e is the eccentricity of an ellipse and 6 is the 
complement of the eccentric angle. By taking a few terms 
of the final series, when e is small, an approximate value of an 
elliptic arc of a quadrant's length is obtained. When e = 
the result is the length of a quadrant of the circumference of 
a circle. 



CHAPTER IX. 

TAYLOR>S THEOREM — EXPANSION OF FUNC- 
TIONS. INDETERMINATE FORMS. 

204. Law of the Mean. — The mean value of / (x) 
between the values / (a) and / (b) is, by Art. 133, 



I f(x)dx 



b — a 

where c is some value between a and h. 

If the function of x is </> (x) = f (x) and Xi is some constant 
value between a and x, then 

X — a X — a 

or f(x)=f(a)+f{x{)(x-a), (1) 

the Law of the Mean, or Theorem of Mean Value. 

If the curve in the figure be the graph oi y = f{x); then 
the ordinate at Pi will be f{xi), the mean value of f{x) 
between /(a) at Pa and /'(a:) at any value x, the integral being 
represented by the area under the curve from x = aiox = x. 

If the curve \sy = f (x), it may be seen that there must be 
at least one point Pi between the points (a, / (a)) and (x, 
f (x)) at which the slope of the tangent is equal to the slope 
of the secant through those points; that is 

^ ^^^^ ~ x-a ~ Ax' 
and hence (1). 

401: 



402 INTEGRAL CALCULUS 

This may be put in the form, 

which may be used to determine increments approximately, 
and is the Theorem of Finite Differences. 




The theorem may be extended so as to express in terms of 
the second derivative the difference made in using the first 
derivative at x = a in place of its value at a; = Xi. Thus, 
if the function of x is (x) = f" (x), and x^ is some constant 
value between a and x, then, 



/"(%) = 



Jj" 



(x) dx 



_ f (^) ~ / (^) /by mean value, V 
X — a X — a \ Art. 133, / 



or f (x)=/(a)+r(x2)(x-a). 

Integrating this equation between the limits x = a and 
X = X, f (a) and /' (X2) being constants, gives 

fix) = /(a) +/' (a) (X - a) +/" (x,) (£^, (2) 
a second Theorem of Mean Value, or the Law of the Mean. 



OTHER FORMS OF THE LAW OF THE MEAN 403 

If the tangent at Pa meet the ordinate MP produced at R, 
then, MR=f (a) + f (a) {x-a); MP = f (x), 
and, therefore, both in sign and in magnitude, 

RP = MP - MR = }" (X2) ^^Z_^'. 

Here the deviation of the curve at P is below the tangent at 
Pa, J" {X2) being negative, and, measured along the line of 

the ordinate MP, is equal to f (X2) - — q"^' ^^^^ ^^® 

curve is above the tangent, f fe) will be positive and RP 
upward. 

205. Other Forms of the Law of the Mean. — The 

theorems (1) and (2) may be given in the following forms. 

In the theorems x, the symbol for the argument in general 
has been used for any value of the argument, a definite value 
but not constant. Now, if Xi be any number between a and 
X, then Xi — a and a; — a are of the same sign whether a is 
less or greater than x; therefore, (xi — a)/ (x — a) is a posi- 
tive proper fraction, 6 say, and Xi = a -{- 6 (x — a) will 
denote any number between a and x. 

Letting x = a -{- h, x — a = h; then theorem (1) will 
become 

f(a + h)=f{a)+ hf (a + dh), (IJ 

and theorem (2) will become 

f{a-\-h)=fia)+ hf (a) + ^f ' (a + d,h). {2a) 

The di of (2a) is not necessarily the same as the 6 of (la). 
If a is replaced by x, the forms become 

f{x + h)=f{x)-{-hf(x + dh), (I5) 

f{x + h)=f(x)+ hf (x) + ^r (x + d,h), (25) 



404 • INTEGRAL CALCULUS 

If a is made zero and then x is put for h, the forms are 

/W=/(0)+*/'(to), _ (1,) 

f(x)=f (0) + xf (0) + p" (9:x). (2c) 

Example 1. — To find the value of 6, if f(x) = x^. Here 
fix) = 2x; f(a + eh) = 2(a + eh); 
and {a + hy = a''-}-2ah-{-h^ = a''-\-h'2{a + ih); 
also, by (la), 

(a + hy =f(a)+ hf (a + dh) = a''-\-h'2 (a-\-eh); 

hence in this case d = ^. 

To find at what point on the parabola y = x^, the tangent 
is parallel to the secant through the points where x = 1 and 
a; = 3. By theorem (1), 

/. xi = 2; or by theorem {la), Xi = a + dh = 1 -\- ^ (2) ^ 2, 
since d = i. 

Example 2. — To find at what point on the curve y = 
sin X, the tangent is parallel to the secant through the points 
where x = 30° and x = 31°. Here 

.,, , sin31°-sin30° 0.51504 - 0.5 ^^ai^T 

f {xO = cosx. = 3io_3QO = 0.01745 ^ ^'^^^^^^ 

.♦. xi = cos-i 0.86177 = 30° 29'; 
hence 2/1 = sin 30° 29' = 0.50729, 6 = f f . 

Example 3. — To show that sin x is less than x but is 
greater than x — \x^. 

f{x) = sinx; f {x) = cos a;; f" ix) = — sinx; 

/(O) = 0; f (0) = 1; r{e,x) = - sin (M- 

• By theorem (Ic), 

sina; = + ic cos (dx), < x, since cos (dx) < 1. 



TAYLOR'S THEOREM 405 

By theorem (2^), 
sin X = + a: — -^ sin (dix) > x — —, since | sin (dix) \ < 1. 

Example 4. — -To show that cos x is greater than 1 ~ J x^. 

f{x) = cosx; f (x) = — sinx; f"{x) = — cos a;; 
/(O) = 1; r (0) = 0; r (Bix) = - cos (M- 

By theorem (2c), 

cos X = 1 — ^x^ cos (01^;) > 1 — J a;^, since | cos (dix) | < 1. 

206. Extended Law of the Mean. — The law of the 
mean or the theorem of mean value in its several forms may 
be used to obtain approximate expressions for a given 
function in the neighborhood of a given point x = a. Still 
closer approximations may be obtained from the law when 
extended in the form of a series arranged according to ascend- 
ing powers oi X — a with the successive derivatives as con- 
stant coefficients. 

For values of x near to a, the higher powers of x — a may 
then become negligible. The most convenient theorem for 
this purpose is the one which follows. 

207. Taylor's Theorem. — Iff{x) is continuous, and has 
derivatives through the nth, in the neighborhood of a given point 
X = a, then, for any value of x in this neighborhood, 

fix) =/(a) +^(s - a) +i^ (x-af+--- 

• • • + £^ (^ - "'"" +"^"^ (^ - ''^"' ^^) 

where X is some unknown quantity between a and x. 
The last term 

Rn{x)=^^-^{x-ar 

is the error in stopping the series with the nth term, the term 
in (x — a)"~i; and the formula is of practical use only when 



406 INTEGRAL CALCULUS 

this error becomes smaller and smaller as the number of 
terms is increased. 

The form of the remainder Rnix) is seen to differ from the 
general term of the series only in that the derivative in the 
coefficient of the power of {x — a) is taken for a; = X instead 
of for X = a. 

The simplest proof of this theorem is the extension by 
integration of the law of the mean — a further extension 
than already used for the theorems of mean value. 

Thus, if the function of x is {x) = f" (x), and X is some 
unknown constant value between a and x, then, 

r /'' {x) dx = f" (X) {x - a) (by mean value, Art. 133) 

Integrating this equation between the limits x = a and 
x = x, 

f (X) - r (a) - f" (a) {x~a)= /'" (X) ^^^, 

[f (a) and f (X) being constants. 
Integrating again, / (a) also being constant, gives 

fix) = f{a) +f (a) {X - a) ^f (a) ^^^ +r (X) ^^ ~ ^^^ 



23 

As this process can be continued to include the nth deriv- 
ative, by induction Taylor's Theorem results. 

208. Another Form of Taylor's Theorem. — If in the 

form (1) X is put for a and {x + h) for x, it becomes ' 

/(a; + /i)=/(x)+/'(x)^+/"(x)|+ • • • , ■ 

where the last term is the remainder after n terms, and B is 
some positive proper fraction. In (1), (a + ^ (x — a)) may 
be used in place of X; and in (2), it becomes (x + 6h) 



the expression, '^ ^^ — {x — a)^, since called by his 



MACLAURIN'S THEOREM 407 

Another form of the remainder called Cauchy's is 

RUx) =/n(a + ^ (X - a)) ^"^ ~ ^]"i\~/^^"' - 

Note. — Taylor's Theorem is the discovery of Dr. Brook 
Taylor, and was first published by him in 1715. He gave it 
as a corollary to a theorem in Finite Differences and there was 
no reference to a remainder. It was Lagrange who, in 1772, 
called attention to its great value and found for the remainder 
f[a + d_^-a)] 
n\ 

name. It has become to be regarded as the most important 
formula in the Calculus. 

The formula known as Maclaurin's Theorem, after Colin 
Maclaurin, was published by him in 1742, but he recognized 
it as a special case of Taylor's Theorem. The two theorems 
are virtually identical as either can be deduced from the 
other. 

209. Maclaurin's Theorem. — If, in the form (1), a is 
made 0; 

/W=/(0)+^-^.+«x^+--- 

where X = 9a; is some unknown quantity between and x. 
Another form of Maclaurin's Theorem is 

where the last term is the remainder after n terms, and 6 is 
some positive proper fraction. 

Cauchy's form of the remainder is 



408 INTEGRAL CALCULUS 

210. Expansion of Functions in Series. — It has been 

shown in examples of Chapter VIII on Infinite Series how 
useful it is to be able to represent a function by means of a 
series. Apart from the purpose of computation such rep- 
resentation may be an aid to an understanding of the proper- 
ties of functions. Taylor's Theorem and Maclaurin's 
Theorem furnish a general method of expanding or develop- 
ing any one of a numerous class of functions into a power 
series. 

For when the error term in Taylor's and in Maclaurin's 
Theorem approaches zero as n increases, each becomes a 
convergent infinite series, called Taylor's series and Mac- 
laurin's series for the given function, about the given point 
X = a. 

Some functions may be expanded by division, some by 
the binomial theorem, others by the logarithmic or the 
exponential series. All of these series are but particular 
cases of Taylor's Theorem. 

211. Another Method of Deriving Taylor's and Mac- 
laurin's Series. — 1. Maclaurin's Series. — If a function 
of a single variable is expanded or developed into a series 
of terms arranged according to the ascending integral powers 
of that variable, and the constant coefficients found, the 
development will be the form of Maclaurin's Theorem with- 
out the remainder. Thus, let / (x) and its successive deriv- 
atives be continuous in the neighborhood of x = 0, say 
from X ^ —a to X = a, and assume that for values of x 
within that interval, 

fix) =A+Bx + Cx'' + Dx'-\-Ex'+ • • • (1) 

If equation (1) is identically true, then the equation 
resulting from differentiating both its members, viz., 

f{x)=B + 2Cx + 3Dx^ + 4:Ex'-\- • • • , 

also is identically true for values of x in some interval that 
includes zero. For similar values of x, the following equa- 



ANOTHER METHOD OF DERIVING SERIES 409 

tions resulting from successive differentiations are identically 
true : 

f'ix) = 2C + 2.3Z)a; + 3-4&2^ .... 

f"{x) = 2.3i) + 2.3-4&+ • . • . 

f^(x) =2-3.4^+ • • • . 

Putting a; = in each of these equations gives : 
A =/(0), B =/'(0), C =m p JllM, E =M, etc. 

Hence, on substituting these values in equation (1), 
/(x)=/(0)+/'(0)x+-M^ + O0^+ ... 

+ ^^+---', ' (2) 

which is Maclaurin's series as in (3), Art. 209, without the 
remainder. 

If fix) is not continuous, no development according to 
powers of x is possible. Thus if J{x) = log x, / (0) = — 00 . 

A power series represents a continuous function, hence no 
power series in x can be expected to develop log x. 

It is evident that, whenever the function or any one of its 
derivatives is discontinuous for x = 0, the function cannot 
be developed in a Maclaurin's series. 

2. Taylor^ s Series. — Let / (x) and its successive deriva- 
tives be continuous in the neighborhood oi x = a, say from 
x = a — h to x = a-\-h, and assume that for values of x 
near x = a, 

f{x)=-A+B(x-a)+C{x-ay + Dix- ay 

+ E(x-ay-\-..., (3) 

is an identically true equation. 

Then the following equations resulting from successive 
differentiations are identically true for values of x near x = a. 



410 INTEGRAL CALCULUS 

fix)=B + 2e{x-a)+SD{x-ay+4:Eix-ay-{-' - • . 
fix) = 2C-\-2-3Dix-a)-\-S-4:E{x-ay-\- • • • . 
f''{x)= +2'SD-\-2'S'4:Eix-a)-{- ' ' ' . 

r{x) = +2.3.4E+ .... 

Putting X = ain each of these equations gives 
A = /(a), B =/'(«), C =ffl Z>=q^, ^=« etc. 

Hence, on substituting these values in equation (3), 
fix) =f{a) +-^ (X - a) +f^ (x-ay+ ■ . ■ 

■•^"^f (x-a)"+ • • • , (4) 



n\ 



which is Taylor's series as in (1), Art. 207, without the 
remainder. 

If in (4) X is put for a and (x + h) for x, it becomes 

f{x + h) =f{x) +/' (x) ^ +/" {x)^_+ ■■■ 

which is Taylor's series as in (2), Art. 208, without the 
remainder. 

Here the development is not according to powers of x, but 
of some value (x — a) orh near to the value x = a. Hence, 
when the values of the function and all its derivatives are 
known or can be found for some one value of x, say a, then 
the value of the function ior x = a -{- h can be found from 
the development. Thus, when /(x)=loga;; / (1) = 0, 

f (1) = 1, r (1) = -1, ni) = 2!, . . . /(")(!) = (-1)"+! 

(n — 1)!; and the series will be 

7,2 7,3 /,4 

iog(i + ;») = /i-|- + |-j+ •••. 

which agrees with (1), Ex. 1, Art. 203. 



EXPANSION BY THEOREMS 411 

212. Expansion by Maclaurin's and Taylor's Theorems. 

— If, on applying to a given function any one of these 
formulas, the last term becomes 0, or approaches as a limit 
when n becomes infinite, the formula develops this function; 
if not, the formula fails for this function. That is, if Rn (x) 
= when n = co , Maclaurin's or Taylor's series is the devel- 
opment of f(x) or f(x + h), respectively. 

If/" (x) increases (or decreases) from/''(0) to/" (x), and the 
sum of the first n terms in Maclaurin's series is taken as the 

x^ 
value oi f{x), the error, being /" (dx) —^, hes between 

/"(O)-. and fHx)-,' 

If /"(x) increases (or decreases) from/"(x) to/"(x + h), and 
the sum of the first n terms in Taylor's series is taken as the 

value oif{x-\-h), the error, being/" {x + dh) —^, lies between 
/»W^ and fix + h)'^,- 



213. Since — . = 



/ytl /y» /y» />• /y '>•'>• 

«*/ »t/ U/ .*/ Uy Uy Uy 



nl 1 2 3 ' ' ' n-2 n- 1 n 



/ytl 

— . = when x is finite and n 



CX) 



for last factor approaches zero. 

Hence, Rn=oo{x) = when /"(^x) is finite, or when 
/" {x + Oh) is finite, in Maclaurin's or Taylor's Theorems, 
respectively. 

214. If / (a;) = f ( — x), the expansion of/ (x) will contain 
only even powers of x; while if / (x) = — / ( — x) the expan- 
sion of/ (x) will involve only odd powers of x. For examples, 
see the expansions of sin x and of cos x following. 



412 INTEGRAL CALCULUS 

215. Examples. — 1. Sin x. — Expansion by Maclaurin^s 
Theorem : 

fix) = sinx, /(O) = sinO = 0, 

r{x) = cosx, /(0) = 1, 

r{x) = -sin a:, r(0) = 0, 

nx) = -cosx, /''(0) = -l, 

^f^{x) = smx, /iv(o)=o, 



/Mx) = sin(x + ^), /nO) = sin(^), 

f^{ex)=sm(dx-^'fj' 

Since sin [ — J is or d= 1 according as n is even or odd, the 

coefficients of the even powers of x will be zero, and only odd 
powers of x will occur, the terms being alternately positive 
and negative. Thus, 

x^ , x^ x"^ , . ^" • //I , ^'"■N 

smx = x-^^ + ^-^^+ . . . +-sm[dx+-j' 

Here, Rn (x) = — ^ sin f ^x + -o" ) > 

x^ 
not numerically greater than —., which has zero for limit; 

/. R (x) = 0. Hence the series 



/y>3 /y»5 /Vi /y»9 

sinx = x-3j + 5|-7| + 9i- ••• (D 

is absolutely convergent for every finite value of x. 

The series converges rapidly and may be used for comput- 
ing the natural sine of any^angle expressed in radians. 

Thus, for the sin (5°43'46''.5 = yV radian), 

sin (0.1 radian) = 0.1 - ^^ + ^^' = 0.09983 .... 

oi o! 



EXAMPLES 413 

180 



For the sin ( 1° = tL radian = 0.017453 . . A 



sinl= = sin(^) = ^-(j^J^, 

+ (iror5V--- =0-017452...; 

/. sin 1°= arc 1°, to five places of decimals. (See Art. 40.) 

2. Cos X. — Expansion may be made by Maclaurin's 
Theorem as is done for the sin x, or the differentiation of the 
sine series term by term gives the series 

/y*^ /y»4 rpx) /y>Cj 

cosx=l-2| + 2n-6! + 8!" ' ' ' ' ^^^ 

which is absolutely convergent for every finite value of x. 

For the cos (5°43'46''.5 = yV radian), 

(0 lY (0 lY 
cos (0.1 radian) = 1-^^ + ^^^ • ♦ - = 0.995007 . . . . 

For the cos ^1° = -|^ radian = 0.017453 . . . ], 

cosi° = cos(^) = 1 - (iioy^+(iioyf! — 

= 0.999847 . . . ; 
/. cos 1° = 1 - 0.00015 . . . 

3. Sin {x + h). — Expansion by Taylor's Theorem. Here 
f{x + h) = sin(x + /i); 

/. / (x) = sin X, f ix) = cos x, f (x) = — sin x, etc. 
Hence, 
sm \x-\-h) — sm x-\-h cos x — ^. sm x — ^ cos x+ — ^ sm x + • • • 

= smx(^l-2j + 4|-gj+ • • ') 

+ cosx(^/i-3j + ^-^+ • • -j (1) 

= sin X cos h + cos x sin h, {h for x in Exs. 1 and 2) 
the well-known relation true for all values of x and h. 



414 INTEGRAL CALCULUS 1 

h^ h^ K^ 
When X = m (1), sin/i = /i — ^ + ^ — ;^+ • • • , 

as in Ex. 1 for x. The series (1) is rapidly convergent for 

TT 1 

small values of h. Thus, let a; =77 and h = -7^ of a radian 

6 100 

= 34' 22^^65; then, 

. /tt ^ 1 V . irL (0.01)2 (0.01)4 X 

+ cosg^0.01--3^ + -^- ...j; 

sin 30° 34' 22^65 = 0.5 (0.99995 . . . )+ 0.86603 . . . 
(0.00999 . . . ) = 0.50863 ... = 0.5 + 0.00863 . . . 

4. Cos {x -\- h). — Expansion may be made in the same 
way as for sin {x -\- h), or the differentiation of the series (1), 
h being constant, gives * 

cos (a; + /i) = cos a; ( 1 - 2^ + ^ - g| + • ♦ • ) 

-smx(^/i-3j+^-yj+ • • • j (2) 

= cos X COS /i — sin X sin h, the well-known relation. 

h? h^ h^ 
When X = in (2), cos /i = 1 — ^^ + t] "" ^ +- * * * » 

as in Ex. 2 for x. 

When x='^ Siiidh = -^oisi radian - 34' 22".65, in (2) ; 
o luU 

then cos 60° 34' 22".65 = 0.5 (0.99995 . . . ) - 0.86603 . . . 

(0.00999 . . . ) = 0.49132 . . . =0.5-0.00868 

5. a^ and e*'. — Expansion by Maclaurin's Theorem. Here 

f{x) = a^, /(0)=a°=l, 

fix) = a- log a, /'(O) = loga, 

/"(x) = a-(loga)2, /"(0) = (loga)2, 

nx) = a^{\ogay, /"'(0) = (loga)^ 



/^ (x) = a=^ (log ay, /" (0) = (log a)^ 

/"(^x) = a^^(loga)^ 



EXAMPLES 415 

.. a^=l+-^—+ 2! + 3! "^ * * • 

Since f"" {x) = a^ (log a)", when a is positive, f(x) and all its 
successive derivatives are continuous for all values of x. 



When X is finite, a'^ is finite. By Art. 213, ^ 7^^ = 0, 



(xloga)'* _^ 
n\ 

when n = 00 and x log a is finite. Hence Rn (x) = when 
n = (X)' and x is finite. Therefore, the exponential series 
xloga (^jogo)^ , (xlogg)"-! , 

is the development of a'' when a is positive and x is finite, 
being then absolutely convergent. 

Value of e^. — Putting a = e in (1), gives (since log e = 1) 

/v» /y>2 ^3 /yi /y«n — 1 

'' = '+T+h+h+h+ ■ ■ ■ +i^+ ■ ■ ■ ■ (2) 

FaZiie o/ e. — Putting a: = 1 in (2), gives 

= 2.718281 .... (See Art. 34.) 

6. Log„(l + A:). — Expansion by Maclaurin's Theorem. 
Here 

/ (a;) = logo (1 + x), / (0) = loga 1 = 0, 

/'W = rf^' /'(0) = «, 

/"W = -(r^2' /"(0) = -m, 



416 INTEGRAL CALCULUS 

/"W = ^^^^7r+iy;^' /"(0) = (-i)"-i(»-i)!m, 

^„(,,)^ (-!);-'(» -!)!>» ; 

/. loga (1 + X) 

/ ^2 0,3 ^.4 fx-dxY /(-1)"~'\ 

= m(x--+ 3--+ . . . +(__j .(^^j. 

When X < — 1, loga (1 + x) has no real value. 
When X = —1, the odd derivatives are discontinuous. 
When X > —l,f(x) and all its successive derivatives are 
continuous. 

Cauchy's form of remainder, 



Rn (X) = f- (dx) 



{n-l)\ 



J. , . (x-BxY (-1)"-! 
gives ^"W = ir+^j --TT^^' 

in which the second factor is finite, and the first factor ■« 0, 
when n = ao and x > —1 and < 1 or a; = 1. 
Hence the logarithmic series 

loga (I+X) 

I X'.X' X'. (-l)n-2^n-l \ 

=n^-2+3-4+"-+ n-1 +"•; (1) 

is the expansion of loga (1 + x) when x > — 1 and < 1 or 
a;= 1. 

Putting —X for X in (1), gives 

(/y»2 /y»3 /y»4 \ 

-x-|-|-|+ • • • j. (2) 

Subtracting (2) from (1), gives 

, 1+x ^ f . a^ . x\ x\ \ /Compare (3), \ .^. 
log" rz^ = 2»(^+ 3 + 5 + 7 + • ■ • j- (ex. 1, Art. 203 j ^^^ 

T ^ 1 . , 1 + X 2; + 1 , .. 

Let X = 7^ — r-^; then :j =- -• (4) 

2z-\- I 1 — X z 



EXAMPLES 417 

Substituting in (3) the values in (4), gives 

••• l<'S.(^+l)=log«.+2™(2^ + 3-^2^+---). (6) 

When 2;>0, 0<x<l; hence the series in (6) is conver- 
gent for all positive values of z. 

When 1 is put for z, loga 2 is found ; then 2 for z^ and loga 3 
is found; thus loga (2: + 1) can be readily computed when 
loga 2; is known. (See Ex. 1, Art. 203.) 

Whenm = 1, a = 6, the Napierian base; thus (5) becomes 

2 + 1 

Dividing (5) by (7) and denoting by iV, gives 

z 

loga iV/log A/" = m\ that is, logaiV = mlogA^. (8) 
Yalue of m. — Putting A/" = a in (8), gives 

1 = mloga; that is, m = 1/loga, (9) 

or N = e, gives, loga e = m; 

.'. 1/loga = logae, or 1 = log a • loga e. (10) 

Value of M. — Denoting the modulus of the common 
system whose base is 10 by M; from (9) and (10), 

^ 0.434294 .... (Compare Art. 38.) 



2.302585 



Note. — When a = 10 and m = M, or when a = e and 
m = 1, all the series in this example become common loga- 
rithmic series, or Napierian logarithmic series, respectively. 



418 INTEGRAL CALCULUS 



EXERCISE XLin. 

2. secx = l+2+-24+;-720]+'--- V''^!<2)- 

/v»3 /y^ /yi /y»9 

3. tan-'x = x-3 + 5-y+g- •••. 

/y»2 /)f3 ^4 

6. e^ = l+x + | + |j + f,+ .... 

/Y»2 /v»3 'y^ 

6. e~^ = 1 — a; + — — ;r-j + — J — • • • , by replacing x by — x in 5. 

p^ p — ^ /v»3 /v»5 

7. sinha:=— 2— =:r + 3-j + ^ + 

e^ J- g— ^ a;2 /p4 

8. cosh X = — 2 — ^-'^ + 2T~'~4T~^*''' ^^ combining terms 

of 5 and 6, or by differentiating terms of 7. 

^ , , sinha; x^ , 2x^ 17 a;' , 

9. tanh x = — 7— = x — -^ -\- -r^ ^-^ + • * • • 

cosh a; 3 15 315 

10. From 5 and (1) and (2) of Examples 1 and 2, making 

"^ X = V— 1 ' X = ix, 

get e*^ = cos X + ^ sin x, (1) 

and e~"^ = cos a; — i sin a;. (2) 

Note. — Putting tt for a; in (1) gives the remarkable relation, e*'^ = — 1 ; 
while putting — tt f or a; in (2) gives e'^^ = 1, hence iir is an imaginary 
value of log 1, the real value being zero. 

11. From (1) and (2) of 10, by subtraction and by addition get 

eix _ e-ix eix _}_ g-ix 
sma: = —^ (3), cos a; = -• (4) 



(by combining termsX 
of 5 and 6. / 



12. Evaluate 



Ce-'^'dx = f' (1 - x2 + ia;4 - ix6 ^ . . . ) ^^, 

Jo »/o 



Get result when end value is 1. When end value is 00 the value of 
the integral is | v^.* This integral is important in the theory of 
probabihty. 

* WilUamson's Integral Calculus, Ex. 4, Art. 116, also Gibson's 
Calculus, Ex. 3, Art. 136. 



THE BINOMIAL THEOREM 419 

216. The Binomial Theorem. — I . The binomial theorem 
is seen to be a special case of Taylor's Theorem by expanding 
(x + /i)"^ in a power series in h by Taylor's formula. Thus, 

f{x-^h) = (x + h)^; :. f (x) = x^, 

f (x) = mx"^-^, f " ix) = m (m - 1) x'^-'^, 

f" {x) = m (m - 1) (m - 2) x"^-^, 



/«-i(x) = m (m - 1) . . . (m - n + 2) x'^-'^+i. 
Substituting these values in Taylor's series (5), of Art. 211, 

{x + h)^ = X- + mx--i h + ^^^~ ^\ -^-^h? 
^ m (m - 1) (m - 2) ^^_,^, ^ ^ ^ ^ 

is the resulting Binomial Theorem. 

Here /'^ (x) = m (m — 1) . . . {m — n -\- \) x""""". 

Hence, / (x) and all its successive derivatives are continu- 
ous for all values of x. 

h'^ii — ey-^ 

Cauchy's form of remainder, Rn (x) =/" {x-\-BK) . _ .^ , 

gives 

P , . _ m (m - 1) . . . (m - n + 1) A - dh V (x + ^/t)"^ 
^"^^^" (^-1)! 'U + W * 1-^ * 

When \x\> h and n = cc , the product of the first and 
second factors = 0, and the last factor is finite; hence, 
R (x) = 0. 

n=oo 

Hence, the binomial theorem holds true when the first 
term of the binomial is greater absolutely than the second. 

When m is a positive integer, the series (1) stops with the 
(m + l)th term, since/" (x) = when n > m, and is therefore 
a finite series of m + 1 terms, 



420 INTEGRAL CALCULUS 

li \h\> X, the expansion may be a power series in x; thus, 

{h + x)"^ = /i- + m/i--i X + ^(^-^1)^""% 2 _!_... 

m(m- 1) . . . (m-n + 2)/^"^--+i __, , 
•••+ (n-1)! "^ +••• ^^^ 

is a true expansion when | /i | > x. 

II. (1 -\- x)"^ may be expanded in a power series in x by 
Maclaurin's Theorem, giving 

/i I N^ 1 I I m(m— 1) „ , m(m— l)(m — 2) „ , 
(l+a;)^=l+mx + — ^^j — Lx^-\ ^ ^ '-x^-{- • • • 

. . . , rnjm- 1) . . . (m - n + 2) 

+ (n-1)! "^ + • ^^^ 

As in case I, when m is a positive integer, the series (3) 
stops with the (m + l)th term and is therefore a finite series 
of m + 1 terms. If m is negative or fractional, the series is 
infinite. The ratio test shows that the infinite series con- 
verges absolutely when | x | < 1 and diverges when \x\> 1 ; 
therefore Rn (x) needs examination only for | x | = 1. 

Here /** (x) = m (m - 1) . . . (m - n + 1) (1 + x)"^-"", 
f"" iSx) = mim - I) . . . (m - n + 1) (1 + Bx)'^-''. 

X" (1 — ^)"~^ 
Cauchy's form of remainder, Rn {x) =/" {Bx) — j ~ — , 

gives 

For values of x between and ±1, the last factor is finite 
for every n; the second factor is always positive and cannot 
exceed unity; the first factor approaches zero as a limit as 
n increases without limit, since it is the expression for the 
nth term of the convergent series 



APPROXIMATION FORMULAS 421 

Hence, R (x) = 0, and the infinite series converges to 

n=oo 

(1 + x)"^ for every value of m, when | a; | < 1. 

For X = ztl, the following results may be found 
proved in Chrystal's Algebra. These cases are not so 
important. 

When X = -\-l, the series converges absolutely if m > 0, 
but ' conditionally if > m > —1, oscillates if m = — 1, 
and diverges if m < —1. 

When X = —1, the series converges absolutely if m > 0, 
and diverges if m < 0. 

If a > 5, (a + 6)"* can be written a"^ (1 + h/a)"" and 
expanded by Maclaurin's formula, since h/a < 1 may take 
the place of x in (1 -i- x)"^. Hence, in this case, 

(a + h)^ =a^ + ma^-^ ^ _^ m (m^ - 1) ^^_, ^, _^ ^ • • , (4) 

which agrees with (1) and is the Binomial Theorem, proved 
true for a > 6 whether m is positive or negative, whole or 
fractional. 

If a < 6, interchange them in (4) and the result will agree 
with (2) and be a true expansion of {b + a)"". 

217. Approximation Formulas. — Often a function may 
be replaced by another having approximately the same 
numerical value but a form better adapted for computation. 
In such cases the given function may be expanded in a series 
and a certain number of terms, beginning with the first, 
taken as an approximate valife of the function; the number 
of the terms taken being according to the precision desired 
for the result. 

The hinomial theorem furnishes one of the most useful of 
the approximation formulas. Thus^ if m denotes a small 
fraction, expanding (1 ± m)'' gives 

(1 zfc m)« = 1 ± nm + "^^^7^^ ^' ± • • • , 



422 INTEGRAL CALCULUS 

where, since m is small, neglecting powers higher than the 
first, the approximate relation, 

(1 6zmY = Idznm (1) 

results. For the special case n = J, 



Vl±m = l±im. (2) 

For h small in comparison with a, the general form is 

V^^±fc = a(l±2^,)- (3) 

For examples: Vl -}- x = 1 -\- ^x — • • • ; 
1 1 . 1 . 1 , 



1+:^ ' Vl+x 2 

For extraction of roots in general, 

1 

(a- ±hr = a(l:±: ~J = a (1 =b x)^, 

7 1 

where x = —. Expanding {1 dcx)"^ gives 



(4) 



^ n n? 2\ n^ 3 ! 

Example. — v^IOOO = ^^1024 - 24 = 4 (1 - yt^)i 
Substituting y|^ for x in the series 

^ ^) - ^ 5 5 j^ 5 10 l5 ' ' * ' 
gives to six figures 0.995268; hence, 

\/l000 = 4 X 0.995268 = 3.981072. 

Since e=^=lH-x+^ + ol+ • • • , when x is small 

e-=l+x (6) 

is the approximate relation. 



APPROXIMATION FORMULAS 423 

From the series for sin x, cos x, and log (1 -\- x), 

sino; = x(l - Jx2), (7) 

cos X = 1 — i x^, (8) 

log(l+a;) =x-ia;2, (9) 

are the approximate relations when x is small. 
When x is small compared with a, 

sin (a it x) = sin a db X cos a, (10) 

log(a + a;) = loga+^- 2^> (11) 

^ =-=F-, + ?-^ (12) 



a zfc X a a^ a^ 
are approximate relations when succeeding terms of the 
respective series are neglected. 

In all these cases the error made in taking the approxima- 
tion for the value of the function may be 
closely estimated from the value of Rn (x), 
the error term, for the particular series em- 
ployed. 

Example. — In considering the length of a 
circular arc and its corresponding chord in 
railway surveying, use may be made of the 
approximate relation (7). Thus, letting s 
denote length of the arc, r the radius, c the 

chord, a the angle in radians ; s = ra and c = 2 r sin 




When a is small, 



a 
2' 



^ . a ^c. q; f-, 1 focY' 
c = 2rsm2 = 2r2[l-g(2J_ 

= ra — ^V ^«^ ; •*• s — c = 2 


kro?, 


4oi4ioU 
thfi prrnr nf tVip nnnrovimflfinn os\.inr\nt pvpp« 


, ra^ 



1920 



424 INTEGRAL CALCULUS 



EXERCISE XLIV. 

1. Expand (x + ?/)'". 2. Expand (x + y^. 

3. Expand 6^+^". 4. Expand log sin (x -f- A). 

5. Expand sin-i (x + A). 6. " Expand e^^^^. 

7. Given / (x) = a;^ - 4 a; + 7, find / (x + 3) and f (x - 2) by 
Taylor's series. Then find / (x + 3) and / (x - 2) by usual algebr<%ic 
method and thus verify results. 

8. Using the approximation formula (12) compute the reciprocal 
of 101; and of 99. Compare results with those obtained by division. 

9. Find the length of the chord of an arc of radius 5729.65 feet 
subtending an angle of 1°: (a) by trigonometric methods; (6) by the 
approximation formula (7). Find results when the radius is 5729.58 
feet. Compare results and find error of approximation. 

10. Find the length along the slope of a road that rises 5 ft. in a 
horizontal distance of 100 ft. by the approximation formula (3). Deter- 
mine to how many places of decimals is the result correct. 

218. Application of Taylor's Theorem to Maxima and 
Minima. — This Article is supplementary to Art. 83, being 
an additional proof of the rule given in that Article for the 
determination of whether a critical value x = a, Sb root of 
f (x) = 0, makes / (x) a maximum, a minimum, or neither. 

Let / (x) be a function of x such that f (a -\- h) and 
f {a — h) can be expanded in Taylor's series, and let / (a) be 
the value to be tested. 

Developing f (a — h) and f (a -{- h) by formula (2), Art. 
208: 

f{a-h)=fia)-hf{a)-\-^^r(a)-^r^^ia)+ • • • 

+ ^f/"(«-W, (1) 
/(a + /i) = /(a) + hf (a) + |/(a) + p''[{a) + • • • 

' ^^^V(« + W, (2) 



in which ^i and 62 are between and 1 in value. 



INDETERMINATE FORMS 425 

When the first n — 1 derivatives of/ (x) are zero for a; = a 
and the nth derivative is not zero for x = a, then, 

f{a-h)-fia) = ^-^f (a - e,h), (3) 

f{a + h)-f{a)=^_fia + e,h). (4) 

Since / (x) and its successive derivatives are assumed to 
be continuous at and near x = a, the signs of /" (a — dih) 
and /" (a + dih), for very small values of h, are the same as 
the sign of /^(a). 

It is manifest that if n is an even integer, / (a) will be a 
maximum or a minimum according as /"(a) is negative or 
positive; and if n is odd, f (a) will be neither a maximum nor 
a minimum wl^ether the sign of /" (a) is negative or positive. 

These conclusions are manifest because when n is even and 
/"(a) is negative, the left members of (3) and (4) are both 
negative, and hence / (a) > f {a — h), / (a) > f {a -\- h); 
that is, / (a) is a maximum. 

When n is 6z;en and /" (a) is positive, the left members are 
both positive, and hence/ (a) < f (a — h),f (a) < f (a + h); 
that is, / (a) is a minimum. 

When n is oc^c?, whether /" (a) is negative or positive, the 
left members have different signs, and hence/ (a) ^/ (a — /^), 
/ (<^) > / (<^ H" ^) ; that is, / (a) is neither a maximum nor a 
minimum regardless of the sign of f"" (a) . 

219. Indeterminate Forms. — It was noted at the end 
of Art. 20 that the derivative oi f{x), 

.Ax=o ^x dx '' ^ '^' 

may be finite, zero, or non-existent, but not 0/0. 

The symbol 0/0 is called an indeterminate form, and 
when / (x) takes that form for some value of x, say a, then 
/ (x) is really undefined for x = a, although it may be defined 
for any other value of x. It is possible, however, that / (x) 



426 INTEGRAL CALCULUS 

may have a definite limit A when x converges to a; it is 

customary then to call / (a) = 0/0 an indeterminate formy 

and to define A as the value of / {x) when x = a, calling it 

the true value of f {x) ior x = a. 

The advantage of having this 'Hrue value" assigned by 

definition is that / {x), being in general continuous, thereby 

becomes continuous up to and including the value a. 

x^ — 4 
Take, for example, the function y = —. For every 

X ^ 

value of X other than x = 2, the function has a definite value, 

4 — 40 

but for a; = 2 it becomes = - . Since the function has 

Z Ji u 

no definite value when x = 2, the limit which the function 

approaches as x converges to the value 2 is assigned as the 

value of the function when x = 2. If 

x = 2 + h, lim^|±4^^^=lim(4 + /i) = 4; 

h=0 Z -h Al — Z h=Q 



lim = 4. 

x=2 X -2 



Thus the true or limiting value of this function which takes 
the indeterminate form 0/0 is 4. 
For values of x other than 2, 

y = - — o = ^ + 2; 



X''-4: l 

X- 2j^=2 



= lim {x + 2) = 4. 



On the graph of y = x -\- 2, the ordinates of points for 
values of x other than 2 represent the values of the function, 
but for X = 2, the function having no definite value may be 
represented by any ordinate lying along the line x = 2. Of 
the values that may be assigned to the function for x = 2, 
there is one value represented by MP = 4, which is the 
limit of the values represented by the ordinates of points on 



INDETERMINATE FORMS 



427 



y = X + 2 SiS X approaches 2; and it is desirable to select 
this value of y as the value of the function when x = 2. 

By this selection the function is defined for a; = 2 and thus 
becomes continuous through that value of the variable x. 




In general, lim / (x) defines the value of the function when 

f{x) is indeterminate for x = a. The expression f {x)]a 
denotes the value of / (x) when x = a. 

; The value of a function of x ior x = a usually means the 
result obtained by substituting a for x in the function. 

When, however, the substitution results in any one of the 
indeterminate forms. 



0/0, 00 /OO, O.OO, 00-00, 0' 



00 



the definition must be enlarged; thus, the value of a function 
for any particular value of its variable is the limit which 
the function approaches when the variable approaches this 
particular value as its limit. 

This definition need be used only when the ordinary 
method of getting the value of the function gives rise to an 
indeterminate form. 



428 INTEGRAL CALCULUS 

220. Evaluation of Indeterminate Forms. — In many- 
cases the limits desired are easily found by simple algebraic 
transformations or by the use of series. When the function 
that assumes the indeterminate form is the quotient of two 
polynomials, or can be put in that form, the following direc- 
tions may be of service. 

f (x) 

1. If the function is of the form vt-^ and becomes 0/0 for 

<f>{x) 

X = 0, divide both numerator and denominator by the lowest 
power of x that occurs in either. If the fraction becomes 
00 /oo for x = 00, divide both terms of the fraction by the 
highest power of x in either. 

f(x) 

2. If the function has the form ) : and becomes 0/0 for 

(f>{x) 

X = a, divide both terms by the highest power of (x — a) 
common to both 

EXAMPLES 

x^ + Sx'^ — 5x1 5 



^1 = 

3 a; Jo 



Sx'- 2a:3 + 6a;|o 6 

When X = 0, this fraction takes the indeterminate form 
0/0. Hence to evaluate it for x = 0, its limit when x = 0, 
must be found. For values of x other than 0, 

x^ + Sx^ - 5x x''-\-Sx- 5 ,~ 



2. 



Sx^- 2x^ + 60; 3x3- 2x2 + 6' 




x3 + 3x2-5x ,. x2 + 3x-5 
iTo 3x4 - 2x' + 6x " iS 3x3 - 2x2 _,_ q 


5 
6 


x3 + 3x2-5x1 
3x4- 2x3 + 6xJ^ 




13 5 

x3 + 3 x2 - 5 X X x2 x\ 




3x4-2x3 + 6x ^ 2,6' 





EVALUATION OF INDETERMINATE FORMS 429 



i^ 3x^-2x3 + 60; ~ i^fi ^ 2 . 6 " 3 ~ 



X X' 



VlH- X - Vl 



Jo 



1. 



X _lo 

By rationalizing numerator, 

Vl-\-x- Vl-x /Vl+x-Vl-x\ iVl+x + Vl-x 



n 



vT+x + vi 



l-x \ 



lim 

x=0 



\/l+x + Vl -X 
Vl +x - Vl - x' 



n=iin 

J x = 



lim -^ 



=]- 



=0 Vl + X + Vl 
4. Vl+x - Vil = 0. 

Joo 

By changing form, 

vr+^~v^=(vr+^-v^) i^^+^ 

( Vl + X + Vx) 
_ 1 + X — X 

~ Vi + x + VJ' 

.-. lim (Vl+x - V^) = lim .— — , — 7= = 0. 
x = oo x=« LVl + X + VxJ 

- X — sin xl _ 1 

By expanding sin x in series, 

X — sin X _ 1 r / x^ ^5 \ -| 



x^ 



1 _x^ x^ 
6 5!^7! 



• • , if X ?^ 0; 

,. X- sinx ,. ri x2 , X* 11 

lim r = lim ' — -I + — — . . . = 7T' 

x=o x^ x=oL6 5!^ 7 J 6 



430 



INTEGRAL CALCULUS 



6. 



smx — X cos 



Jo 3 



x" Jo 3 

By expanding sin x and cos x in series, 
sin x — X cos 



x' 



- = ^[(" 



0^ x^ _ 

3! "^5!" 



^4! 



)]■ 



lim 

x=o L 



sm X — X cos 



"] = '; 



lim 





_x'\ 



= lim 

x=0 



30 



30 

+ 



4- 



)] 

1 
3* 



221. Method of the Calculus. — For the form 0/0, to 
which all other indeterminate forms may be reduced, Tay- 
lor's Theorem furnishes a general method of evaluation. 

I. When/(x) and </> {x) are continuous functions of x and 

fix) 

■ ; i reduces to the form 0/0 for x 

<l>{x) 



a, the value of 
lim ^-^ is desired. 

That is, if the ratio of two functions of x takes the form 0/0 
when X = a, then the ratio of these functions when x = a is 
equal to the ratio of their derivatives when x = a. 

li f{x -\- h) and ^{x ^ h) can be expanded by Taylor's 
formula in the neighborhood oi x = a, it is seen that 

h=Ql4>{a-\-h)\ <l>'{a)' L</>WJ« \J> {x)]a 

By Taylor's Theorem [Art. 208, formula (2)], putting 
a for X, 

f{a + h) ^ f{a)+hf{a + dih) ^ f (a + d,h) 
<t>{a-\-h) <t> (a) + h(j)' (a + Oih) 4>' {a + ^s/i) ' 
since /(a) = 0, <A(a) = 0; [See also (!«), Art. 205] 



METHOD OF THE CALCULUS 431 

h=o U (a + ^) J h=o U (« + WA 4> {o) 
lif{a) and <t>' ip) are both zero, then [(2a), Art. 205] 



i'S U (« + /») J ™ U" (a + «4/i) J 0" (a: 



In this way it is seen that if, for x = a,f (x) and <t> (x) and 
their successive derivatives, including their nth derivatives, 
are zero, while /"+' (a) and (^"+' (a) are not both zero, then 






(2) 



f (x) 
If the function . . takes the form 0/0 when x is infinite, 

<t>{x) 

by putting x = - the problem is reduced to the evaluation 
z 

of the limit for 2; = 0, and hence the method applies to this 

case also. 

fix) 
II. Form GO /oo . — When the function ^^— -r takes the form 

0(x) 

cc/qc, it can be reduced to the form 0/0, by writing it in 

the form —y-z / tt^- This form can be evaluated as before. 
4>{x)/ f{x) 

Thus, let / (a) = 00 and (a) = 00 , a being finite or in- 
finite; to show T^l =[^1. 

Now ~rr^ = — 7-^ / jy-r , which is in the form 0/0. Ap- 

0(a) <f>(a)/ f{a)' 

plying formula (1), 



432 INTEGRAL CALCULUS 

*'(a) 






[/(a)? 



U(x)J<. r^iMl U'wJ" 



If ^^V^ is indeterminate, continue according to formula (2) 

(a) 
until two derivatives are obtained whose ratio is deter- 
minate, which ratio is the limiting value sought for the 
function. 

III. Other Forms. — The evaluation of the other indeter- 
minate forms may be made to depend upon the preceding. 

(a) Form • oo . — When a function f {x) ' <l> {x) takes the 
form • 00 for X = a, it may be reduced to the form 0/0 or 
cxd/oo ; thus, 

}{x)'(t>{x)='-Y' or -^• 



<i>{x) Six) 

ih) Form oo — oo . — By some transformation and simpli- 
fication, a function taking the form oo — oo may be reduced 
to a definite value, or to one of the preceding indeterminate 
forms. 

(c) Forms 0°, oo°, 1°°. — These forms arise from a func- 
tion of the form [f {x)Y'^'^\ This function may be reduced 
to the form %. Thus let y = [} (a;)]'^(^\ whence 

log2/ = <^(a;).log[/(x)]. (3) 

Since for each of the given forms, (3) takes the form O.oo, 
the evaluation is effected as in (a), the value of y being 
found from log y. 



METHOD OF THE CALCULUS 433 

EXAMPLES. 



1. 5-- 

X 



^1- 

a; — sin x~] _ 1 
^' ~^' Jo~6* 
X — sin x] _ . . a; — sin x~| _ 1 — cos x^ 
x^ Jo ~ ' •** x3 Jo ~ 3x2 J^ 

_ sin x~| _ cos a:~| _ 1 
6 X Jo 6 Jo 6 

3. = na''-^. 

X — a ]a 

^n _ ^n-| Q x"" — a"1 nx'^^n , 

X — a Ja X — a Ja 1 Ja 

^ a^ — 6^1 , a 

4. = logr- 

X jo o 

= n J • • = log a • a^ - log • 6* 

a: Jo x Jo * Jo 



a^ - 6^" 



= log a — log b = Jog ^- 



6. e--e---2x 
X — sm x Jo 



]. 

10, . e^- e-^- 2x 1 _ e^ -f g-^ - 2 1 
Jo ' * * X — sin X Jo 1 — cos X Jc 

Jo cosx Jo 



X — sin X Jo U ' X — sin X Jo 1 — cos x Jo 



sinx 



« "°«('+i)L*»- 



434 



INTEGRAL CALCULUS 



log(n-^)'] =a;log(l+^)] = a (by Ex. 6); 

•■• i'^l)']?'- ■•■ (■4)1 -■ CD 

.*. (1 + x)^ = e. (Compare Cor., Art. 34.) 

-1 1 

9. (1 - x)^ = - or e-^. 
Jo e 

log(l - xy] = i-log(l - x)l = :^1 = -L 
Jo X Jo 1 — ^Jo 



or 



EXERCISE XLV. 

Evaluate the following indeterminate forms: 

^ 1 — cos x~ \ 1 j^ a; — 1 ~| _ 1 



4 



1. 



I. sec X — tan x L = 0. 

- tan X — sin a;~j _ 1 
sin^x Jo 2 

4. (sinx)*^^^1_ 

5. x^^^^l = 1. 

Jo 

6. sin X log x]o = 0. 

7. a;^]o = 1. 

8. (l+x^)^^ = l. 



10. {^y\-n^' 

a; — IJi 

12. ^^^] = 2. 
sm re Jo 

^« x — sin~^ x ~\ _ 1 

^^- ""^^ Jo -^6" 



sm-^o; 
tan x — X 



14. 

X — sm a; 

15. (loga;no = L 



!-• 



16. 



a;i-^J,=-e-i. 



EVALUATION OF DERIVATIVES 



435 



222. Evaluation of Derivatives of Implicit Functions. — 

1. Find the slope of x" - a'xy + bY = at (0,0). Here 

dy 
dx 



Hence 



dy 
dx 



1 =4^^^1 =% (Art. 105) 

Jo.o 



12x2 



dy 
dx 



dx lo.O " _ 9 7,2 ^ 

dx 



dx 



0,0 



2Vp 
dx_ 



0,0 



(„._26.^)+a^^l =0; whence 1^1 =0or^ 
V dx) dx^Qfi axjo.o o- 



2. Find the slope of x^ - 3 axy + 1/^ = at (0,0). 

Arts. -Y-\ = or 00 . 



CHAPTER X. 

DIFFERENTIAL EQUATIONS. APPLICATIONS. 
CENTRAL FORCES. 

223. Differential Equations. — A differential equation 
is one that involves one or more differentials or derivatives. 
An ordinary differential equation is a differential equation 
which involves one independent variable only. The deriv- 
atives in such an equation are therefore ordinary derivatives. 

The order of a differential equation is the order of the 
highest differential or derivative which it contains. 

The degree of a differential equation is that of the highest 
power of the highest differential or derivative which it con- 
tains, after the equation is freed from fractions and radicals. 

For examples: 

dy = m dx, (1) 

are ordinary differential equations. Equation (1) is of the 
first order and first degree, (2) is of the second order and first 
degree, and (3) is of the second order and second degree, 
after being rationalized. 

224. Solution of Differential Equations. — It has been 
seen in foregoing chapters how when an equation expressing 
a functional relation between two variables is given, the 
differentiation of the equation gives a differential equation 
expressing the rate of the function. On the other hand, it 
has been seen that the rate of a function being given a differ- 

436 



COMPLETE INTEGRAL 437 

ential equation is thereby formed, the integration of which 
yields the function, indeterminate though it may in general 
be. 

It is this finding of the function from an equation involving 
the derivative that constitutes the solving of a differential 
equation. 

The general solution of a differential equation is the most 
general equation free from differentials or derivatives, from 
which the given equation may be derived by differentiation. 

The general solution, for example, of the equation 



J=C, is 2/ = Cx + C 



Ij 



and of ■T^ = 0, is y = Cx + Ci, also. 

Thus, when -~ represents the slope and j\ the flexion, 

this function is any non-vertical Une in the plane. Here, 
y = Cx, y = Ci, y = Oy y = X, y = 2 X -\- I, . . .are par- 
ticular solutions, which are included in the general solution. 
(See Ex. 1, Art. 115.) 

[Note. — The general solution of a differential equation 
may not include all possible solutions. A solution not in- 
cluded in the general solution is called a singular solution. 
The discussion of such solutions is beyond the scope of this 
book.] 

The general solution of a differential equation of the nth 
order contains n arbitrary constants of integration (for ex- 
amples, see Arts. 140, 141, 161) ; to determine these constants, 
n conditions connecting the function, the variable, and the 
successive derivatives must be known. The general solution 
is called the complete integral or primitive of the differential 
equation. 

225. Complete Integral. — When a differential equation 
is given, passing by integration to the complete integral is 



438 INTEGRAL CALCULUS 

solving the equation. It has been proved that every differ- 
ential equation has a complete integral, and that when the 
equation is of the nth order the integral contains n arbitrary- 
constants. The complete integral then contains one, two, 
... or n constants that do not appear in the differential 
equation, when that equation is of the first, second, ... or 
nth order. If the complete integral be differentiated these 
constants are eliminated, whatever may be their particular 
values, hence they are called arbitrary. 
Example 1. — Let a given equation be 



differentiating, h ^ ~ ^ ^^^ 

differentiating again, ^ = - = -^ (from (2)); (3) 

••• -S-| = 0, ((2) Art. 223) (4) 

is the differential equation. 

To find a general form for the complete integral of (4), 

dy d^y dTYi 

solve by letting m = -^; then -7^ = -j-; substituting in 

... . dm „ dm dx 

(4) , gives X-. m = 0, or — = — : 

^ ^' ^ dx m X 

integrating, log m = log x + log c = log ex, 

where c is a constant; 

/. m = ex, or -p = ex; (2) 



integrating again gives 

2/=|x2 + ci, (10 

the complete integral, in which c and Ci are the arbitrary 
constants; and which has the form of equation (1). 
Whatever be the value of Ci, equation (!') represents a 



COMPLETE INTEGRAL 439 

2 

parabola on the ?/-axis with latus rectum - ; hence (20 is the 

c 

differential equation of all such parabolas. 

Equation (4) is the differential equation of all parabolas 
whose axes are on the 2/-axis. 

Suppose (4) is given, and the problem is to find a function 
y that shall satisfy that equation, have its first derivative 
equal to 1/p when x = 1, and be equal to k when x = 0, 
These conditions give, from (!') and (2'), 
k = + Ci; l/p = c, 
and hence the function, 

y = xy2 v + k. (Compare Ex. 2, Art. 161.) (1) 

In this way the constants can be determined when the 
necessary conditions are known. 

N.B. — Equations of the second order with one variable 

fd^v dv \ 
■j^, -7^ , X J = 0, may often be solved by the 

method used in this example. (Art. 232, IIL) 
Example 2. — Let the equation be 

g + 26g+(62 + co^)i/ = 0. ((4) Art. 54.) (1) 

It is seen in Art. 54 that this differential equation results 
from the differentiation of the equation 
y = ae~^' sin {o)t — a), or y = e~^' {A sin cot-\- B cos coQ . (2) 

To show that (2) is the solution of an equation of the 



(3) 



form of (1), let 


, y = e-^^u and (1) becomes 




d'^u . „ ^ 


where 


co2 = 62 + co2-(i.26)2; 


then by Ex. 3, 


following, 




u = Asinojt + B cos mt, 


and 


y = e-^^{A sin o)t -{- B cos oot). 


(Compare Art. 


233, III.) 



(2') 



dt 



440 INTEGRAL CALCULUS 

Example 3. — Let the equation be 

^ +0)2^ = 0. ((3) of Ex. 2.) 
Multiplying by 2 du gives 

integrating, 

(^J = -0,2 (1^2 4_ (7^) =. ^2 (^2 _ ^2)^ taking Ci = -aS 

extracting root, '17 '^ ^ ^^ ~ ^' 

Integrating, /^^.^ =/, 

gives sin~i - = coi + C2, 

or, solving for w, 

w = a sin (coi + C2) = A sin coi + B cos co^, 
where A = a cos C2 and B = a sin C2 are arbitrary constants. 

Example 4. — Let the equation be 
dH 2 „ 

Multiplying by 2 di^ gives 

integrating, f-^j = co2(?^2 + (7^); 

extracting root, 17 ~ ^ ^'^ ~^ ^i* 

Integrating, / , = / co di, 

gives log {u + Vi^2 + Ci) = coi + C2, 

or, solving for it, w = Ae"^ + -Be~*^*, 



THE NEED AND FRUITFULNESS 441 

where 2 A = e^ and 2 B = — Cie~^ are arbitrary constants. 
By means of the hyperbohc functions this result may be 
written in the form 

u = a sinh (ot) + b cosh (coQ, 

where h-\-a = 2A and b — a = 2B. 

Hence, in this case, a solution of (1) of Ex. 2 is 

Example 5. — Let the equation be 

resulting from (3) of Ex. 2, when 6^ + co^ = 6^, or co^ = 0. 

du 
Integrating gives -j- = Ci,u = Cit -{- C2. 

Hence in this case, the solution of equation (1) of Ex. 2, is 

where Ci and C2 are constants. 

Note. — The foregoing examples are solutions of important 
differential equations, that of Ex. 2, as shown in Art. 54, 
being the typical form for damped vibrations. 

226. The Need and Fruitfulness of the Solution of 
Differential Equations. — Attention has heretofore been 
called to the need of finding the inverse of a rate, in solving 
many problems that arise in everyday life as well as in science. 
In fact the inverse problem is more often the real question 
demanding solution. It has been shown (Ex. 5, Art. 115) 
how, when the acceleration, the rate of change of the speed 
of a moving body, is known, the velocity and the distance for 
any time are found by the solving of a differential equation. 
(Ex. 1, Art. 161.) 

It has been shown (Art. 42), that when a function has the 
general form y = ae^^, the rate of change is proportional to 
the function itself, and that so many changes in Nature 



442 INTEGRAL CALCULUS 

occur in this way that the law of change, known as the 
Compound Interest Law, is also called the Law of Organic 
Growth. Now, if it is known that some function changes at 
a rate proportional to itself, expressing this by the differen- 
tial equation, 

-^ = ky, or kdx = —; 
dx ^' 2/ 

then, kx = I — = loge y + c, or y = e^^-^ = Ce^^, 

*J y 

where C = e""^ is an arbitrary constant. 

The only function whose rate of change is proportional to 
itself is thus shown to be of the form Ce^^^ (or ae^^), where C 
and fc (or a and 6) are arbitrary, and k (or h) is the factor 
of proportionahty. This may be expressed also by the 
statement, that the only function, whose relative rate of 
change (logarithmic derivative) is constant, is Ce^^ (or ae^^). 

It has been shown (Art. 73), that when a point has simple 
harmonic motion its relative acceleration is a negative con- 
stant. Thus, when the displacement of a point is given by 
the equation y = asm (oit — a) , there results the differential 

d^y 
equation -^ = —^'^y, where co is constant, and hence the 

relative acceleration is -p / 2/ = — co^. 

Conversely, when the motion, as in a vibration, is due to 
a force that increases with the distance from the central 
position, the acceleration, being according to Newton's 
second law of motion proportional to the force, is 

dh ,„ 

where, as the force acts towards the origin, the acceleration 
is negative when s is positive and positive when s is negative. 



THE NEED AND FRUITFULNESS 443 

From the relation v dv = atds, gotten by eliminating dt in 
dv/dt — at and ds/dt = v; 

Cvdv= I - kh ds; .: v^ = Ci- fcV; 

putting Ci = k^a^, 

y = ^=:kV^r^^; f /' = fkdt; 

dt ' J Va^-s^ J 

whence . sin-^ (-) = /c^ + C2 

or s = a sin (kt + C2) = A sin kt -\- B cos kt, 

where A = a cos C2 and 5 = a sin C2 are arbitrary con- 
stants. This equation for s is the characteristic equation of 
simple harmonic motion; the amplitude of the motion is a, 
the period is 2 w/k, and the phase is — C2//C. 

Thus, it is found that, when the acceleration along a 
straight line is a negative constant times the distance from 
a fixed point, the only motion resulting is the simple har- 
monic motion. 

In general, it has been shown that, whenever the rate of 
change of a function of a single independent variable is 
known and also the value of the function for some one value 
of the variable, it is possible to find by integration the value 
of the function for any value of the variable. 

Hence it has followed that the solution of a differential 
equation gives the area under any curve whose equation is 
known, thus solving the problem that had baffled the 
mathematicians of the ages before the discovery of this 
general method of effecting the quadrature of curves of any 
degree. 

When it is recalled that the magnitude of any quantity 
whatever, whether of volume, mass, weight, force, work, 
etc., may be represented by an area under a curve, the fruit- 
fulness of the solution of many differential equations is 
recognized. 



444 INTEGRAL CALCULUS 

Note. — In this chapter and in the foregoing chapters the 
differential equations that have been solved have been for 
the most part ordinary, involving the function and one 
independent variable. While a general discussion of differ- 
ential equations, including those other than ordinary, is too 
large a subject for a first course in the Calculus and is beyond 
the scope of this book, some of the special equations are so 
important that their solution has been given, and some more 
will follow. In Art. Ill, the solution of some differential 
equations that have the form M dx + N dy = was effected, 
and in Art. 112, the definition of an exact differential equa- 
tion of that form was given. 

227. Equations of the Form Mdx + Ndy = 0. — In this 
form M and N are functions of x and y. The variables are 
said to be separated when M, or the coefficient of dx, contains 
x only, and N contains y only. 

When M dx -\- N dy is an exact differential (as defined 
in Art. Ill), the total differential of some function of x and 
2/, then 

Mdx + Ndy = 0, (1) 

is an exact differential equation. After applying the test 

l^=dx' «6) Art. Ill), 

and finding the condition satisfied, integrate the coefficient 
of dx regarding y as constant, putting 



fMdx+f{y), (2) 



and then determining / (y) so that 

du 



. =N. (3) 

dy 



Or regarding x first as constant, put 

u = fNdy+f(x\ (4) 



EQUATIONS 445 



and so determine / (x) that 




dx 


(5) 


These equations involve the conditions, 




,^ du ,r du 
dx dy 


(6) 


Example. — Solve (3 a;^ + 4 xy) dx + {2x^-\-2 y) dy = 


= 0. 


-— - = 4aj = — — , hence, the condition is satisfied. 
dy dx 





J{3x^ + 4:xy)dx +f{y) = x^ + 2x'y +f(y); 

f'{y) = 2x^ + 2y, 
f'{y)=2y and f(y)=y'; 



^=2x'+f(y) = 2x^ + 2y, 



hence 

u = x^ -\-2 x^y + 2/^; •*• x^ + 2 x'^y -\- y'^ = C 

is a solution and is the complete integral. 
When the equation, 

M dx -\- N dy = 0, 

is not exact, it may be multiplied by some factor that will 
make it exact in some cases. This factor is called an inte- 
grating factor. Rules have been given for finding an in- 
tegrating factor, but in many cases a factor, or several 
factors, that will make the equation become exact, may be 
found by inspection. For Example, see Ex. 2, Note, Art. HI, 
ydx — xdy = 

is made an exact differential equation by either the factor 
^-2^ y-2^ Qj. {xy)~^, and a solution effected in each case. 
For another example, 

(1 -]rxy)ydx-\-{\- xy)xdy = 0, 
ydx-\-xdy -\- xy^ dx — x^y dy = 0, 
or d {xy) + xy^ dx — x^y dy = 0; 



446 INTEGRAL CALCULUS 

dividing by x^'if' gives 

d{xy) dx dy ^ ^^ 
{xyY X y ' 

1 X - 

h log - = log c, ox X = cye'y. 

xy ^y *^ ' 

228. Variables Separable. — An equation of the form 

Mdx-\-Ndy = 0, 

in which the variables are separated can be solved by in- 
tegrating its terms separately. The variables are separable 
when the equation can be put in the form 
J{x)dx + F{y)dy = Q. 

Example 1. — cosxdx — sin ydy = 0. 
Here sin x -jr cosy = C is evidently the general solution. 

Example 2. — Va^ — y^dx -}- Va^ — x^dy = 0. 

Dividing by V^iF^^ \/W^^^ ; . ^^ + . ^^ = ; 

V a^ — x^ V a^ — 2/2 

X V 

integrating, sin"^ - + sin"^ - = C ; 

a a ^ 

taking sine of each member, the first member being a sum, 
gives X Va^ — y"^ -\-y Va^ — x^ = a^ sin C = d. 
Example 3. — (1 — x) dy — {1 -\- y) dx = 0. 

Dividing by (l-x)(l+2/); T^-T^ = ^\ 

X -\- y i X 

integrating, 

log (1 + 2/) + log (1 - x) = log c or Ci, 
or (1 -\-y) (1 — x) = c or e^K 

The final equation may be gotten at once by inspection. 

229. Equations Homogeneous in x and y. — If an equa- 
tion is homogeneous in x and y, the substitution of 

y =^ vx 



EQUATIONS HOMOGENEOUS IN X AND Y 447 

will give a differential equation in which the variables are 
easily separable. 

Example 1. — Solve (x^ + y^) dx — 2xydy = 0. 

Putting y = vx and dividing by x^ gives 

(1 -\- v^) dx — 2 V {x dv -{- V dx) = 0; 

separating the variables, _ ^ = 0; 

integrating, log lx{l — v^)] = \ogc; 

putting y/x for v, the solution becomes 
x2 — 2/2 = ex. 

Example 2. — Solve {x^ + y^) dy — xydx = 0. 
Putting y = vx and dividing by x^ gives 
(1 + v"^) {xdv -\- V dx) — vdx = 0; 

separating the variables, — | f- ^ = ; 

' V X v^ 

integrating, log v + log x — ^v'"^ = C = log c; 

V V x^ 

putting - for v gives log- = -^t—^j 
X c ^ y 

or y = ce^y\ 

Example 3. — Find the system of curves at any point of 

which as {x, y), the subtangent is equal to the sum of x and y. 

dx 
From the conditions, the subtangent being y-r-y 



hence, 
dividing by y 





dx 






''dy^ 


'■ x + y. 


ydx- 


xdy = 


■ydy; 


2 ydx — 


xdy 


dy 


'. ■ y. 




"J' 



X 

integrating, - = log ?/ + log Ci = log Ciy, 



or y = cey 

is the general equation of the system of curves. 



448 INTEGRAL CALCULUS 

230. Linear Equations of the First Order. — A " linear 
differential equation is one in which the dependent variable 
and its differentials appear only in the first degree. 

The form of the linear equation of the first order is 

dy + Pydx = Q dx, (1) 

where P and Q are functions of x or are constants. 

The linear equation occurs very frequently. The solution 

of dy -{- Pydx = 0, or dy/y -\- P dx = 0, 

is log ?/ + log e-^ "" = log c, or ye^ "^ = c. 

Differentiating the latter form gives 

ey'^'\dy + Pydx) = 0, 

which shows that e-^ ^ is an integrating factor of (1). Mul- 
tiplying (1) by this factor gives 

e^ ^ (dy -{- Py dx) = e^ ^Qdx; 
and this, on integration, gives 

/'''=fe/'''Qdx. (2) 



ye-' 



The equality expressed in (2) may be used as a formula 
for solving any linear equation in the general form (1). 
Example 1. — Solve x dy — y dx — x^ dx = 0. 
Putting it in the general form (1), it becomes 

dy — -dx = x^ dx. 
^ X 

Hence, i P dx = — j — = — log a: = log - ; 

/Pdx log- 1 

X 

and / e-/ '^Qdx=jxdx = — -\-C. 



EQUATIONS OF ORDERS ABOVE THE FIRST 449 

Substituting these values in formula (2) gives 

| = ix2 + C, or y = ^x'-\-Cx. 

Example 2. — Solve dy -\-ydx = e~^ dx. y = (x -\- C) e~*. 
Example 3. — Solve cos X' dy -\- y sin x- dx = dx. 

y = smx-\-C cos x. 
Example 4. — Solve (1 + x^) dy — yxdx = a dx. 

y = ax + C Vl + x^. 

Th a 

Example 5. — Solve dy -\--ydx = —dx. x'^y = ax-\-C. 

231. Equations of the First Order and nth Degree. — An 

equation of the first order and nth degree, which is resolvable 

into n equivalent rational equations of the first degree, may 

be solved by the solution of the equivalent equations. To 

dt/ 
illustrate, let p = -7^ in the examples. 

Example 1. — Solve ^^Y + (x -\- y)^ + xy = 0. 

The given equation is p^ -{- (x -\- y) p -\- xy = 0; factoring, 
(V + 2/) (P + ^) = 0) which is equivalent to the equations, 
p -\- y = 0, p -{- X = 0, oi which the solutions are, 
\ogy + x + C = 0, 2y + x^ + 2C = 0. 
The combined solution is 

(log?/ + x + C)(2y + x'-\-2C)=0. 
Example 2. — p^- ax^ = 0. 25 {y + C)^ = 4 axK 

Example 3. — p^ - 5 p + 6 = 0. 

(y-2x-C){y-Sx-C) =0. 
Example 4. — p^ _ ax^ = 0. 343 {y + Cy = 27 d^. 

Example 5. — p^ + 2 xp'^ — y^p"^ — 2 xy^p = 0. 

(y -C)(y-\-x'- C) {xy + C|/ + 1) = 0. 

232. Equations of Orders above the First. — Examples 
will be given of solving four special forms of such equations. 



450 INTEGRAL CALCULUS 

I. Equations of the form -r-^ = f{x). 

The solutions of equations of this type can be gotten by 
n successive integrations. Examples have already be6n 
given in Art. 161 and in other Articles. 

Example. — d^y = x^ dx\ 2/ = i^ H — ^ H — ^ + C^x + d. 

d^ti 

II. Equations of the form j-^ = fiy)- 

For these equations 2 dy is an integrating factor. 

Example. — Solve ^ + a^^ = 0. (1) 

Multiplying by 2 dy gives 

integrating, 

/^y = -ay + Ci = a2 (ci^ - 1/2), where Ci = aV. (2) 

From (2), dy/Vci" - y'' = adx; (3) 

integrating (3) sin-^ y/ci = ax -{■ C2, (4) 

or y = Ci sin {ax + 02), (5) 

which may be written, 

y = Asinax -\- B cos ax. (6) 

See Ex. 3, Art. 225, where this solution was obtained, and 
in Art. 226, the equation for simple harmonic motion results; 
and there, for the differential equation, 

dh _ , ^ 

df ~ ^ ^' 
2 ds might be used as an integrating factor. 

(d^y dy \ 

-j-^, . . . , -7^, xj = 0; that 

is, equations of the nth order with y absent. 



EQUATIONS OF ORDERS ABOVE THE FIRST 451 

The solution of a differential equation of this form was 
given in Ex. 1, Art. 225. 

p + — ^y +u d^ _dp d/^ _ d^'-'^p 

^"^^ ^~dx' ^^^"^ dx' ~ dx' ' ' ' ' dx-~ ~d^^' 

Substituting these values in the general form gives 

J d--^p dp \ „ ... 

which is an equation of the {n — l)th order between p and x. 
Example 1. — To show that the circle is the curve for 
which the expression for radius of curvature is constant. 



[-(i)T_ 



d^ 
dx' 

Substituting p for dy/dx, inverting the fractions, and sepa- 
rating the variables, gives 

dp _ dx ^ 

integrating, -^^=±^, 

a being arbitrary constant; 

1 . - dy , X — a 

solvmg tor p, p = -f- = zL 



dx Vr^ - {x- ay 

whence y — b = zL VR"^ — (x — ay, 

h being arbitrary constant; hence, {x — ay -{- {y — by = R^j 
for all circles of radius R. 

When n is 2, the equation being of the second order, the 
substitution 

_ dy dp _ d^y 
dx' dx dx^' 
reduces the equation given to an equation of the first order 
in dp/dx, p, x. Solving, if possible, gives a relation of the 



452 INTEGRAL CALCULUS 

form / {p, X, C) = 0. This is still of the first order, in x and 
y, and may be integrated. 

d}s 1 1 
Example 2. — Solve ;772 + T + 72 "= ^• 

Putting p = ^ gives ^ + - + ^ = 0. 

Separating variables, — dp = —^ — dt] 

integrating, — P = — t + log ^ + Ci. 

Integrating again, 

8 = log^ - t\ogt + (1 - Ci)t - C2, 

which gives in the case of motion the relation between the 
space or distance and the time. 

(d^v\ dii \ 

dx^)' ' ' ' ' dx' V^^' 

Put «-^- then^-^^, ^-^2^ i^/W ... 
i'ut p - ^^, then ^^, - p ^^, ^^3 - p ^y2-^P[^y) , etc. 

Substituting these values in the general form gives an 
equation of the (n — l)th order between p and y. 

Thus when n is 2, the equation being of the second order, 
the substitution 

_ dy dp _ d^y 
^~dx' '^d^'d^^ 

reduces the given equation to an equation of the first order 

in y and p. This is solved, if possible; and then dy/dx put 

for p, giving an equation in x and y of the first order to be 

integrated ior y. 

dh 
Example. — Solve at = j^ = f{s). 

Put p = „ = - then ^=^ = /W, 

giving the known relation vdv = at ds. 



LINEAR EQUATIONS OF THE SECOND ORDER 453 

Integrating gives — = I f{s)ds + C, the energy integral, 
called so from the relation 

the equation of kinetic energy, where Fs is work done by a 
force F through a distance s. 

When / (s) is given, v is replaced by ds/dt and the integra- 
tion of the resulting equation gives the solution in terms of s 
and t and the equation may be solved for s. 

Special examples under this case were given in II and in 
Art. 225, Exs. 3 and 4. 

233. Linear Equations of the Second Order. — The 
general form of the equation of the second order is 

where P, Q, R are functions of x alone or constants. 

The complete integral of all linear equations is the sum of 
two functions, called the complementary function and the 
particular integral. 

The complementary function is the complete integral of 
the equation when R, the term independent of y and its 
derivatives, is zero. This function will contain two arbitrary 
constants, when the equation is of the second order. 

The particular integral is any solution of the equation as 
it is in the general form, and contains no arbitrary constant. 

To consider the complementary function in which P, Q 
are constants, let the equation be 

I. Let y = e^^, k constant; then substituting in (1) gives 

(/b2 -{-ak + b) e^^ = 0. 
If fc is a root of the quadratic equation, 

fc^ + a/c + 6 = 0, (2) 



454 INTEGRAL CALCULUS 

called the auxiliary equation, e** will satisfy (1). The two 
roots ki, /c2 of (2) are 

and e^'i^, e^^^ are two solutions of (1). Hence the complete 
integral of (1) is 



where u = v J a^ — 5. For special cases: 

II. If a^ = 4 b, equation (2) has two equal roots, h = k2 = 
— ^ a. In this case (3) becomes 

y = {A +5)e-^«^, 

where (A + B) might be replaced by one constant C. When 
a^ = 4 6, let y = e'^^'^u and (1) becomes, without the 
factor e-2«^, 

dx ^' 

of which the complete integral is u = A -\- Bx. 

Examples of this solution have been given in Arts. 224 and 
225. The complete integral of (1) when the auxihary equa- 
tion has two equal roots, each — J a, is 

y = {A -^ Bx) e-^*^^. (4) 

III. If a^ < 4 b, the roots of (2) are imaginary. Again, 
let y = e~^"^ u and equation (1) becomes 

g + m^« = 0, (5) 

where | a^ — 6 = — m^ and m is real. Now (5) is satisfied 
hy u = cos mx, u — sin mx ; its complete integral is then 

u = A cos mx + B sin mx, 
and therefore the complete integral of (1), when a^ < 4 6, is 
y = e~^^^u = g-^ax (^^ cos mx + B sin ma;). (6) 

To show how (3) and (6) are written when the roots of (2) 



LINEAR EQUATIONS OF THE SECOND ORDER 455 

are known: when the roots of (2) are real, let i a^ — 6 = n^; 
the roots are then, —ia-\-n,—^a — n; and the solution is 

y = g-iax (^gnx _j_ Be-''''); 

when the roots of (2) are imaginary, let J a^ — 6 = — n^, 
and the roots are then, —\a-\-ni, —^ a — ni; and the 
solution is 

y = Q-'^ax (^^ (>Qg Yix -\- B sin nx)y 

so that instead of e"''^, e~"^^, there are cos nx, sin nx. 

It may be noted that the auxiliary equation is written by 

putting k^ for -^ , k ior -^, and omitting y. 

The solving of a linear equation of the second order in- 
cluding these three cases has been given in Art. 225, Ex- 
amples 2, 3, 4, 5; for the equation for damped vibrations. 

Example 1. — Solve ^ + 8^ + 25 2/ = 0. 

The auxiliary equation is 

/c2 + 8 /c + 25 = 0, 

and its roots are /ci = — 4 + 3 ^, /c2 = — 4 — 3 i. Hence, 
the complete integral is 

y = e-4a; (A COS 3 a: + B sin 3 x). 
Example 2. — Solve 

^ - 2 ^ - 35 2/ = 0. fc2 - 2 /b - 35 = 0. 
dx^ dx 

The roots of the auxiliary equation are 

ki = -5, /C2 = 7. 

Hence, the complete integral is 

y = Ae-^^ + Be^^ = e (Ae-^^ + Be^^). 



456 INTEGRAL CALCULUS 

APPLICATIONS. 

234. Rectilinear Motion. — I. When the acceleration is 
constant, 

da dh ^ dh ds 

dt = de = ^' dF = "' " = d^ = «' + "°: 

s = ^af + Vot + So. 

For bodies falling freely towards the earth from moderate 
heights, the acceleration g being taken constant, 

V = vo — gt, s = Vot- i gt^ + So; 
from rest, 

v= -gt, s= -\gt\ (Ex. 5, Art. 115.) 

Projected outward from rest, h = v^t — \ gt^ (Ex. 6, Art. 

116), where h is height from point of projection. 

II. When the acceleration varies as the distance. Let 

dh 
a = -775 = —ks, where /c, a constant, is the acceleration at 

a unit's distance from the origin ; and let the body be of unit 
mass at an initial distance r; then, 

/dsV 
t;2 = (-77 1 = Ci — ks"^ = kr^ — ks^, where kr^ = Ci; 

V = -r- = 0Vr^ — s^; 
at 

t = r- f /^ = k-^ cos-1 - (+ C2 = 0), 

t = 0, when s = r; 
whence 

s = r cos (kn) = r sin I kH -{--]) 

where the last result is gotten if the positive sign of the radical 
is taken in integrating. 



RECTILINEAR MOTION 457 

Putting s = 0, gives v = r Vk, the velocity at the origin, 
and t = 7r/2 k~^f | Trk'"^, f irk'"^, . . . , or s = r, gives t = 0, 
27r/A;2, 4 7r//c2 , . . . Hence the motion is periodic, the 
period being 2Tr/k^, which is independent of the initial 
distance. 

Differentiating s = r cos (k^t), gives 

'^ — ~j1 — ~ ^^'* sin ik^i) , 

which expresses the velocity in terms of the time that the 
body has been moving. 

It is seen that this is a case of simple harmonic motion. 
(Arts. 73 and 226.) 

It has been shown in Art. 190, Cor., that the attraction of 
a sphere of uniform density for an internal particle varies as 
the distance from the center. Hence, a particular case of 
the periodic motion just considered would be that of a body 
which could pass freely through the earth, taken as a homo- 
geneous sphere. Such a body would vibrate through the 
center from surface to surface. To find the time of this 
half period : 

t = Trk~^ = 3.1416 V20900000/32.17 sec. 
= 42 min. about. 

III. When the acceleration varies inversely as the square 
of the distance. 
Let k be the acceleration at unit distance from origin; 

dh k 

then, a = ^=--,, 

where s is to be taken always positive; multiplying by 2 ds, 
integrating, ^ = (|J = 2fc g - J), (1) 



458 INTEGRAL CALCULUS 

where s = r, when v = 0, which gives the velocity of a 
particle at any distance s. 
For the time, 

2fc ,, —sds 

— at = . 1 

negative, since s decreases as t increases, 

- ["1 r-2s _ r 1 "I 
|_2 Vrs — s^ 2 Vrs — s^J 

integrating between limits corresponding to t = t and 
t = 0, gives 

When the particle arrives at the origin, s = 0, therefore, the 
time to the origin from the point where s = r is 

t = — 

It is seen from (1) that the velocity = when s = r, and 
= 00 when s = 0; hence the particle approaches the origin 
with increasing velocity. While the attractive force causing 
the acceleration is very great near the origin, there can be 
no attraction at the origin itself; therefore, the particle 
must pass through the origin; and the conditions being the 
same on either side of the origin the motion must be retarded 
as rapidly as it was accelerated; hence, the particle will go 
to a point at a distance r equal to that from which it started 
and the motion will continue oscillatory. 

An illustration of this general case has been given in 
Art. 193, where the attraction of the Earth for an external 
particle was considered as the cause of motion. 

IV. When the acceleration varies as the distance and the 
motion is away from the origin. 



RECTILINEAR MOTION 459 

Let k again be the acceleration per unit mass at unit 
distance from the origin; then 

a - ^^2 - ics; 



''' ■g) = 2..<^a; 



using 2 ds as an integrating factor gives 

2d^ 

dt 

fdsV 
integrating, v^ = i-rrj = ks"^ -\- Vq^, 

where Vq is the initial velocity; whence, 

s = -^, (e^'^ - 6-^^0. (See Ex. 4, Art. 225.) 
2 k"^ 

Here, as t increases s also increases, and the particle 
recedes further and further from the origin ; and the velocity 
also increases and becomes oo when s = t = oo. Thus in 
this case the motion is not oscillatory. 

V. When the acceleration is constant and the motion is 
in a medium whose resistance varies as the square of the 
velocity. 

In Art. 195 the case where the motion was towards the 
Earth has been given. Let now the particle be projected 
outward with a given velocity vq. Using again gf/c^, as the 
coefficient of resistance, the resistance of the air on a particle 
for a unit of velocity, and taking the particle of unit mass, 
with g constant, 

g=-.-..^g; (1) 

whence, ttt^ = —kgdt; 

integrating, tan-MA;-,-j = tan-^ (kvo) — kgt, 



460 



INTEGRAL CALCULUS 



where C = tan"^ (kvo) ; solving, 



V = 



ds _ 1^ kvp — tan kgt 

dt~ k' 1 -{-kvotsin kgt' ^^^ 

which gives the velocity in terms of the time. To get it in 
terms of the space; from (1), 



±M1 



-2gk^ds; 



integrating, 



log 



^+K'-:J 



1 + kW 
where c = log (1 + A;W); whence, 



= -2 



-=(fj= 



1 



v,^e-^9k^s _ ^^ (1 



g-2 gk^s\ ^ 



Writing tan kgt in (2) in terms of sine and cosine and 
integrating, 

s = T^ log {kvo sin kgt + cos kgt), 
K g 

which gives the space described by the particle in terms of 

the time. 

235. Curvilinear Motion. — Let a body slide without 

friction down any curve ah. The 
acceleration caused by gravity 
at any point P is ^ sin a, where 
a = PTD, PT being a tangent 
to the curve. 

Let PT = ds; then -PD = 
dy; hence. 




d's 



= gsma — 



n^y 



(1) 



d^2 ^— ^ds 

Let t/o be the ordinate of the initial point on the curve; then 
V = when y = yo. 



SIMPLE CIRCULAR PENDULUM 



461 



Integrating (1) gives 



^=Jl = ^2g(2/o-2/). 



(2) 



It follows from (2) that the velocity of a body acquired 
by moving freely down any frictionless path is the same, 
and is what it would acquire in falling freely through the 
vertical height between the initial and terminal points. 

/ds 
— , the time will depend upon the path. 

236. Simple Circular Pendulum. — Consider the motion 
of a particle on a smooth circular arc under the action 
of gravity as the only force. 




— Y 



Taking the axis of x as vertical, the equation of the circle is 

\f- — 2ax — x^. (1) 

Let K {h,k) be the point where the particle starts from rest, 
and P {x, y) where it is at the time t. Then the particle will 
have fallen through the height h — x, and hence from (2), 
Art. 235, 



ds 



. = ^ = V2,(/» 



x). 



(2) 



It is seen from (2) that the velocity is a maximum when 
X = and a minimum when x = h; so the particle will pass 



462 INTEGRAL CALCULUS 

through following the curve to the point K' where x = h 
and will oscillate between K and K'. 

To find the time in passing from K to K' ', from (2), 

^l — ——=== , negative since s decreases as t increases, 

V 2 g{h — x) 

— adx „ /-.NT adx 

from (1) ds = -—' (3) 



V2g{h- x) (2 ax - x^) 



While this expression does not admit of direct integration, 
approximate values of the integral can be gotten as shown 
in Ex. 6, Art. 203. When the arc is small, the approximate 
value of the time can be gotten from (3) thus : 



T=2t= -~ r 

V2a.A 



adx 



h V(/i — x)x{2a — x) 
a r^ dx 



, by taking {2 a — x) = 2a, 



' aJo Vhx 

If instead of moving on a curve, the particle is assumed to 
be suspended by a rod or cord of no weight, it becomes a 
simple pendulum, existing in theory. 

To reduce (3) to the form known as ^'an elliptic integral" 
of the first kind; let h = a vers a, x — a vers 6, a = KCOj 
being constant and 6 variable; then 

h — X = a (vers a — vers 6) = a (cos 6 — cos a) ; 
dx = a sin 6 dO; and (2 ax — x'^) = a^ sin^ d. 

Substituting in (3) : 



_ _J_ f" 2 g ' g sin ^ d(9 

'\/2g Jo Va (cos ^ — cos a) a^ sin^ Q 

-ill 



a T" dd 



9 Jo \/sin2a/2- sin2^/2 



SIMPLE CIRCULAR PENDULUM 463 



i'r 



= v^X 



Jo 2 V \sin a/2/ 

2 sin q:/2 cos d0 



S' Jo sin a/2 V 1 - sin^ (^ V 1 - sin^ a/2 sin^ 

= 2 V - / . -, by cancellation. (5) 

^ S' Jo Vl - sin2Q;/2sin2(/,' 

Here is defined by sin <j> = ^ — 4^, giving by differenti- 

sin a/ ^ 

ation, 

, ,^ cose/2dd/2 

COS0rf0 = r^ j^ , 

sm q;/2 
whence, 

,2 sin a/2 cos cjidcj) _ 2 sin a /2 cos <^ d^ 
cos ^/2 ~ a/i - sin2a/2sin2 

For change of hmits, 6 = a when = 7r/2, and = 
when = 0. 

Putting k = sin a/2 in (5), it becomes 



•^ 1 

(1 - /c2 sin2 <f>)-^ d(t> 

g Jq 



=Vs['+©'''+6-l)'"+fi11)"'-+ ]■(«> 

The form / {I — ¥■ sin^ 0)~^ c?</) is ''an elliptic integral 

of the first kind.'\ 

When a, and hence k, is small, only the first two terms of 
the final series will give a close approximation to the value 

of T, and the first term alone gives the value tt y - , the expres- 

y 



464 



INTEGRAL CALCULUS 



sion usually taken as the time of a vibration; 27ry - being 

the value taken for a complete oscillation back and forth. 

237. Cycloidal Pendulum. — A particle moves along the 
arc of a cycloid; find the time of descent. 




From (2) of Art. 235, 



dt 



= V2g{yo-y). 



(1) 



The equation of the cycloid referred to OX and OF is 

X = a vers~^ y/a + a/2 ay — y^. 
Hence, ds = —V2a/y dy, which substituted in (1), gives 



^ gJy Vyoy-y^ ^ { 



vers" 



-i?y~f\. 
yojy ' 



i/^L_vers-i^l 
^ ^L 2/0 J 



Voy-y' 

:. t=Tv\J-, when i/ = 0, and 2^=T = 2xy-, 

the time of one oscillation of a pendulum if it swings in the 
arc of a cycloid. 

The time of an oscillation being independent of the length 
of the arc, the cycloidal pendulum is isochronal. 

The pendulum is described in Art. 97, Ex. 3. 

The cycloid is the curve of quickest descent, the Brachysto- 
chrone; that is, the curve down which without friction 
gravity will cause a particle to fall in the shortest time. 



CYCLOIDAL PENDULUM 



465 



The following is a comparison between the times down the 
chord of a circular arc, a circular arc, and the arc of a cycloid. 
For the time down the chord I of a circle 



r 



2rx-x^; t = 2\/-- 



From (1), Art. 235, ^^ = gdna = g—, from the circle, 



dt^ 



ds gl 



^ = ^ = 27^ (+C = 0, since v = 0, when t = 0), 
5 = 1^ ^' (+C = 0, since s = 0, when ^ = 0), 



= v/^=V>hen. = L 



c 


V 


^ 


\\r 


^ 


^v. 


B 


TTa ^V 


(' 


^ P^^ 


\: 


.j^^^ 



For the time down the circular arc A0\ 



(6^\ 



236/ 

= ^V^ [1+0.07+0.01+ . . . ]=0.547ry/-=(L7...)\/-. 
For the time down the arc of a cycloid from A to 0; 



4 Y 

2-T-r-- = 0.539 



Vr(i-6--.)v1^ 



least. 



466 



INTEGRAL CALCULUS 



From the figure, 



r = — - — a OY a = 



4r 



7r2 + 4 
hence k^ = - or P = (2r.2a). 

Hence, Vr ^''^ * * V^ <(l-7 • • V^ < 2 V^- 
It is at once evident that 



9 



9 



9 ^9 

It may be seen that the approxiniate value 

|s/r = (1.57... )v/r< (1.6... )y/r. 

238. The Centrifugal Railway. — The centrifugal rail- 
way is an example of a simple circular pendulum where the 
cord of suspension is replaced by a track. 

Neglecting the resistance of 
friction and of the air, the forces 
acting on the car are the force of 
gravity and the normal reaction 
of the track. If h' is the diam- 
eter of the circular track and h 
the height from which the car 
starts from rest, find the relation 
between h and h', so that the car 
will make a complete revolution without leaving the track. 
The centrifugal reaction at the highest point of the track 
must be great enough to overbalance the weight of the car. 
The velocity at 5 is i; = V2gh; Sit T, v = V2g {h- h')) 

Wv^ W 
whence, ^^72 = ^^72"^^^''"'''^ = ^' 

giving 4: {h — h') = h'; hence h = ^ h\ to balance; and, 




PATH OF A LIQUID JET 467 

therefore, h should be greater than | h' , for the car to com- 
plete the revolution without leaving the track. 

239. Path of a Liquid Jet. — If a small orifice be made 
in the vertical side of a vessel containing a liquid like water, 
and a short tube be inserted so as to direct the current 
obliquely, horizontally, or vertically upward, the velocity of 
efflux will be the same, since the pressure of fluids at the 
same depth is the same in every direction. To find this 
velocity, let v be the velocity, w the weight of the liquid 
issuing with that velocity per second, and h the head or 
height of the surface of the liquid above the orifice; then 
the equation for energy is 

wh = - — v% 
2 g ' 

in which wh is the work w can do in falling through h, and 

1 w 

- — y2 is stored up energy in w as it issues from the orifice. 

Supposing no loss of energy, they are equal; hence, 

v^ = 2gh or v = V2gh; (1) 

that is, the velocity of efflux is the same as that of a body which 
has fallen freely through the height h. 

Now each particle of liquid issuing from the orifice will 
have the same velocity and will follow the same path. The 
path, when the tube is not vertical, will be a parabola whose 
directrix is fixed in the surface of the liquid supposed to be 
kept at a constant level by more liquid entering the vessel 
(Art. 196). If the Hquid issue obhquely, its equation is 
given in (3), Art. 196. If the liquid issue horizontally, o: = 0, 
and the equation becomes 

x' = ~y = 4:hy. (2) 

y 

The equation (2) may be derived thus : let a particle issue 



468 



INTEGRAL CALCULUS 



from the orifice with a velocity v, and in t seconds be at a 

point P; then 

X = vt (distance, v constant), 
y = iot^ (freely falling body). 

Eliminating t between these equations gives 

„ 2v^ 




If the X and y of any point of the jet is measured, the equa- 
tion (2) can be used to determine the actual velocity of flow 
from the orifice. If this is done, the coefficient of velocity 
is given by 

actual velocity 



Cv = 



theoretical velocity 



the actual velocity being less than the theoretical on account 
of friction at the edge of the orifice. 

The path is derived without taking into account the 
resistance of the air, as when the path is in a vacuum. 

When the orifice is at the center of the vertical side of a 
vessel kept constantly full of liquid, it can be easily shown 
that the horizontal range of the jet is a maximum and equal 



DISCHARGE FROM AN ORIFICE 



469 



to 2 h, the height of the vessel; and at equal distances above 
and below the center the range will be the same. 

The coefficient of velocity for a small sharp edge orifice is 
0.98; and, for a short tube, it is about 0.82. 

240. Discharge from an Orifice. — If a is the area of^a 
small orifice, then the theoretical discharge, or the quantity 
of liquid issuing in a unit of time, is 

Q = av = a V'2gh. (1) 

On account of the contraction of the jet at the orifice and the 
diminution of the velocity the actual discharge is 



Q = CcCvav = 0.6 a V2 gh, 



(2) 



where Cd = CcCv = 0.62 X 0.98 = 0.6 about, for a standard 
orifice; for a standard short tube, Cd = 0.82, the coefficient 
of contraction being unity. 



K 


V 




1 




f 


y. 




'^ 






< 


--* -> 





For a small orifice the head is taken as constant and as 
that on the center, and for heads greater than twice the 
height of the orifice that gives the discharge almost exactly. 
For large orifices under low heads the variation of head over 
the orifice, causing a variation in the velocity of the jet and 
therefore in the discharge, makes the formulas above in- 
apphcable for exact results. 

Let h be the head on the center of a rectangular orifice of 
breadth h and depth d; and let the rectangle be supposed 



470 INTEGRAL CALCULUS 

to be divided into horizontal strips of area h Ax, x being the 
distance from the center Hne of the rectangle. The quantity 

d 

Q = lim y;5 Ax V2g{h-x) = b V2g T {h - x)^ dx; 

Ax=0 ^ J _d 

2 
d 

= 5 V2^ f (l - ^ -^, - . . . )dx (.by.^-P-^idA 
J d\ 2h Sh^ I Vmg m series / 

= 6,V2-^(l-g-|l-2^f^...). (3) 

It is seen that the quantity in the parenthesis is less than 
unity, and the discharge is therefore less than that given 
by (1). 

For h = 2 d, the value of the parenthesis factor is 0.997, 
so for heads greater than twice the height of the rectangle 
the discharge may be figured from Q = caV2 gh, where c is 
the coefficient of discharge. 

Integrating without expanding (h — x)^ gives 



d 



Q = hV2g T {h - x)Ux = hV2g{ - Uh - x)il 

-hHhtf-i'-tf} 

If the orifice extends to the surface and the bottom is h 
below, 

Q = hV2~g^-Uh- x)t J = f hh V2^, (5) 

which is just f the quantity that would flow through an orifice 
of equal area placed horizontally at the depth h, the vessel being 
kept constantly full. 

The mean velocity Vm is seen in (5) to be | V2 gh. 

For any vertical orifice formed by a plane curve whose 
vertex is at the depth hi below the surface of the liquid in 



DISCHARGE FROM AN ORIFICE 



471 



a vessel of height h, kept constantly full, the formula for dis- 
charge is 

Q = J'"'^ 2 y V2g{h + x) dx. (6) 

To get the time of emptying the vessel; let the surface be 
z below the top at the end of the time 
t, z = when ^ = 0; then the quantity 
discharged in an element of time is 

dQ = 2 V2g I \ Vx-\-hi- zdx dt, 

z being constant during this integration; 

and since in the same time the quantity 

discharged through the orifice must be 

A dz, A being the area of the section of the vessel at depth 

z, it follows that 



(7) 





zdx 



Example. — Water is flowing from an orifice in the side 
of a cylindrical tank whose cross section is 100 sq. ft. 
The velocity of the jet is a/2 gx, x being the height of the 
surface above the orifice; and the cross section of the jet is 
0.01 sq. ft. Find the time it will take for the water to fall 
from 100 ft. to 81 ft. above the orifice. 

For this example the formula becomes 

A r^i _i 

t = 7= I X ^ dx (where x is height of surface 

caV2gJh 

above orifice and a is area of the orifice) 



100 



r 



"2<ix = 



10000 



0.01 V2 g J 100 8 

= 2500 (10 - 9) = 2500 sec. = 41f 



2/iM 
Ji 



/taking\ 
[g = 32/ 



mm. 



472 INTEGRAL CALCULUS 

CENTRAL FORCES. 

241. Definitions. — A central force is one which acts 
directly towards or from a fixed point and is called an attrac- 
tive or a repulsive force according as its action on any particle 
is attraction or repulsion. The fixed point is called the 
center. The intensity of the force is some function of its 
distance from the center. 

The path of the particle is called its orbit. All the forces 
of Nature that are known are central forces. 

242. Force Variable and Not in the Direction of Motion. 
— Let a particle of unit mass be projected in any direction 

and acted on by an attractive 
force F. The path will be in 
the plane passing through the 
center of force and the line of 
projection. 

In this plane let 0, the center 

^ of attraction, be the origin and 

the pole, and let (x, y) or (p, 8) be the position of the par- 
ticle P at the time t. The equations of motion are, from 
the components of F parallel to the axes OX and OY, 

^=-Fcosd=-F-, ^=-FBme= -F^- (1) 
dt^ p dv p 

Multiplying the first by y and the second by x and sub- 
tracting, 

d^y d^x ^ /». 

integrating, ^~i~^'£^^' ^^^ 

where his o, constant. 

From X = p cos 9 and y = psin 6, 

dx = cos 6 dp — p sin 6 dd, 
dy = sin dp + p cos 6 dd, (4) 



FORCE VARIABLE 



473 



which in (3) gives 



de 
dt 



(5) 



Multiplying equations (1) by 2 da; and 2 dy and adding: 
2dx^x_±2^dy_d^ ^ _ 2F {xdx -\- ydy) , 
dt' P ' 



■■■ <&h<M}> -^*' 



(6) 






+ P^'f^=-2Fdp, by (4); 



[?©"+?]=-¥*. ^^<«- (') 



Putting p = - and hence dp = ^, (7) becomes 

XL U 

dm+u 



(du^ , ^ 2F, 



whence 



d'^u , 



F 
-— - = 



(8) 



which is the differential equation of the path; and as the force 
F will be given in terms of p, and therefore in terms of u, the 
integral of the equation will be the polar equation of the path. 

Let the central attraction vary 
inversely as the square of the dis- 
tance; to find the path. 

Let the particle be projected 
from the point Pq with a velocity 
V, R the value of p for the point 
Po, /3 the angle between R and the 
line of projection; and let K be 
attraction for the unit mass at unit distance, and t = when 
the particle is projected. Then since the perpendicular from 
the origin to the tangent is 




P 



dd 



[Art. 77, (9)], the velocity -n = -, from (5); 

ai 7j 



474 INTEGRAL CALCULUS 

and as at Po, p = R sin jS, 

F=^^; h=VRsm^. 
As the force varies inversely as the square of the distance, 

77" 1 

F = ^ = Ku^, where u = -; 

p2 p 

(Pu K 

hence, dF'^^^ ¥' ^^^^ ^^^' ^^'^ 

Using 2 du as integrating factor and integrating, 

when i = 0, ^ = - = ^, and (^^j +^^ = 7^, by v^ = -, 

and Art. 77 (9); 

V^ 2K _ Vm-2K 

' h' hm h'R ' 

substituting this gives 

^ ^ ^ Vm-2K 2Ku 

Hence, 

which shows that the velocity is greatest when p is least, and 
least when p is greatest. 

Changing the form of (9) to 

dv? Vm-2K , K^ IK V /1AN 

+ TT - (^p - ^^j ' (10) 



and simphfying by letting -^ — h, and 7^^ h tt = c^, 

fi It U hr 

gives 



[c^ - (m - 6)2]* 



FORCE VARIABLE 475 

the negative sign of the radical being taken. Integrating 

gives 

_.u — h . , 

cos ^ = d — c, 

c 

where c' is an arbitrary constant; 

.*. u = b-i-c cos {d - c'). (11) 

Replacing in (11) the values of h and c and the value of h, 
and dividing both terms of the second member by K, gives 

P=i = BWHin^m 

which is the polar equation of the path and is the equation 
of a conic section, the pole being at the focus, and the angle 
{d — c') being measured from the shorter length of the major 
axis. For if e is the eccentricity of a conic section, p the 
focal radius vector, and cj) the angle between p and that point 
of a conic section which is nearest the focus, then 

p = - = -^ '-, or -^ '-' (13) 

u 1 + e cos (/) I -\- e cos 4> 

Comparing (12) and (13), it is seen that, 

e^ = 1/K^ {V^R -2K) RV^ sin^ ^8 + 1 ; (14) 

<l> = e - c'. (15) 

Since the conic section is an ellipse, parabola, or hyper- 
bola, according as e is less than, equal to, or greater than 
unity, and from (14), e is thus, according as V^R — 2K\i 
negative, zero, or positive; therefore, it is seen that 

V^ < -^-, e < I, and the orbit is an eUipse, 

2 K 
V^ = —f^, e = 1, and the orbit is a parabola, 
K 

2 K 

V^ > -^ , e > 1, and the orbit is a hyperbola. 



476 INTEGRAL CALCULUS 

Corollary 1. — By (1) of Art. 234, III, it is seen that the 
square of the velocity of a particle faUing through an infinite 
distance to a point R distant from the center of force is under 
the law of attraction now considered 2K/R. Hence the 
conditions above may be expressed by stating that the orbit, 
described about this center of force, will be an ellipse, a 
parabola, or a hyperbola, according as the velocity of pro- 
jection is less than, equal to, or greater than the velocity 
through an infinite distance. 

Corollary 2. — From (15) it is seen that ^ — c' is the angle 

between the focal radius vector p, and that part of the 

principal axis which is between the focus and the point of 

the orbit which is nearest the focus; that is, it is the angle 

POA in the figure; and hence, if the principal axis is the 

initial line, c' = 0. 

2 K 
Corollary 3. — If the orbit is an elhpse, V^ < —^ ; hence, 

£1 

e2 = 1 - 1/^2 (2 X _ v^R) RV^ sin2 ^, from (14) . (16) 

The polar equation of the ellipse is 

^ g (1 - e^) . 
1 + e cos </) ' 

comparing it with (12), corresponding terms give 

RWsin^^, 



a (1 - e') = 



K 



substituting for 1 — e^ its value from (16) and solving for a; 

KR /17\ 

" = 2K-V'R' ^^^' 

which shows that the major axis is independent of the direction 
of projection. 

In the figure, Po is the point of projection; FPo = R; PqT 
is the line along which the particle is projected with velocity 
V; FPoT = ^, the angle of projection; FP = p; PFA = d; 



KEPLER'S LAWS OF PLANETARY MOTION 477 

PT = p = Rsin^; if i3 = 90°, the particle is projected from 
an apse, A or A', an end of the major axis. 

du 
To determine the apsidal distances, FA and FA^; -^ = 0; 

du 

. 2KU.2K V^ _ f ... ,,., 

•• "-^^ + Ffl-¥ = °'^''°"^(^)' (^^) 

the two roots of which are the reciprocals of the two apsidal 
distances, a(l — e) and a (1 + e). 

Since the coefficient of the second term of (18) is the smn 
of the roots with their signs changed, 

^ +r7TV7V = ^; .-. a{l-e^) = ^, (19) 



a(l-e) ' .a(l+e) ¥ ' ^ ' K 

which is one-half the latus rectum. 

From (5) p^ dd = h dt; and area swept over by the radius 
vector is 

which shows that the areas swept over by the radius vector in 
different times are proportional to the times, and equal areas 
will be described in equal times, li t = 1, A = ^ h; hence, 
h = twice the sectorial area described in a unit of time. 
For the time of describing the ellipse, calling the time, T; 

rp _ 2 area of ellipse _ 2Trah 
~ h ~~h~ 

= 2 7ra^Vn^ (f,om (19)) 
^Ka (1 - e2) 



= 2^1- 



g_' (20) 

which is the periodic time. 

243. Kepler*s Laws of Planetary Motion. — From a long 
series of observations of the planets, especially of Mars, 
Kepler deduced the following three laws which completely 
describe planetary motion. 



478 INTEGRAL CALCULUS 

I. The orbits of the planets are elUpses, of which the sun 
occupies a focus. 

II. The radius vector of each planet describes equal areas 
in equal times. 

III. The squares of the periodic times of the planets are 
as the cubes of the major axes of their orbits. 

The statement of these laws marked an epoch in the 
development of mechanics, for the investigations of Newton 
as to the nature of the attractive force led to his discovery of 
the law of universal gravitation. The conclusions deduced 
by Newton from Kepler's three laws will be briefly shown. 

244. Nature of the Force which Acts upon the Planets. — 
(1) From the second of Kepler's Laws, it follows that the 

planets are retained in their orbits 
by an attraction tending towards 
the Sun. 

Let (x, y) be the position of a 
planet at the time t, referred to 
rectangular axes through the Sun 
in the plane of the motion of 
the planet; X, 7, the component 
accelerations due to the attraction acting on it, resolved 
parallel to the axes; then the equations of motion are, 

^ = Y ^ = y- 

By Kepler's second law, if A be the area described by the 
radius vector, dA /dt is constant, 

•■• 'i - \'''ft=\ (from (5), Art. 242) 




=^ o\^-^ — y-n) = ^ constant, from 
Z\ dt at/ (3)^ Art. 242. 



FORCE WHICH ACTS UPON THE PLANETS 479 

Differentiating gives 





d'x 


■0; 






. xY 


-yX = 


(from 


(1)) 


) 


• 


X 

' Y 


X 

■ -, or 

y 


y = 


X 



which shows that the axial components of the acceleration, 
due to the attraction acting on the planet, are proportional 
to the coordinates of the planet; and therefore by the 
parallelogram of forces, the resultant of X and Y passes 
through the origin. Hence, the forces acting on the planets 
all pass through the Sun's center. 

(2) From the first of Kepler's laws it follows that the 
central attraction varies inversely as the square of the 
distance. 

The polar equation of the ellipse, referred to its focus, is 

a(l-e^) 1 l+ecosB 

P = r~i n or - = u= —jz ^> 

1 + e cos p a (1 — ^2) 

which by differentiation gives, 

fu ^ 1 

d^2 + ^ a (1 - e") ' 

and, therefore, if ¥ is the attraction to the focus, by (8), 
Art. 242, 

a(l -62) p2' 

Hence, ij the orbit be an ellipse, described about a center of at- 
traction at the focus, the law of intensity is that of the inverse 
square of the distance. 

(3) From the third law it follows that the attraction of 
the Sun (supposed fixed) which acts on a unit of mass of each 



480. INTEGRAL CALCULUS 

of the planets, is the same for each planet at the same 
distance. 

By Art. 242 (20), ^ = ^a\ 

Since by the third law, T^ varies as a^, K must be constant; 
that is, the strength of the attraction of the Sun must be the 
same for all the planets. Hence, not only is the law of force 
the same for all the planets, but the absolute force is the same. 

The third law shows also that the law of the intensity of 
the force is that of the inverse square of the distance. 

Since the planets move in ellipses slightly different from 
circles, assume for simplicity that their orbits are actually 
circles. If Ri, R^, Rs are the radii and Ti, T2, T3, the respec- 
tive times of revolution of the planets, Kepler's third law 
may be written as follows : 

Ri' R2' R,' , , 

^=^=^= ••• =a constant. 

The expression for the central acceleration of motion in 
a circle is a = v^R = 4:Tr^R/T^, or T^ = Air^R/a (Art. 70 (2)). 
Substituting this value gives 
aiRi^ = a2R<i' = azRz' = constant; or a = constant /i^^. 

Note. — Arts. 242, 243, 244, are based on the discussion 
in Bowser's Analytic Mechanics. For a fuller discussion, see 
Tait and Steele's Dynamics of a Particle, and Percival Frost's 
translation of Newton's Principia, Sec. I, II, III. 

245. Newton's Verification. — The greatest of Newton's 
achievements is considered an achievement of the imagina- 
tioUj his conception of the universaHty of natural law. At an 
early age he (in 1666) conceived with his far-reaching mind 
the then daring idea that the sublime, inscrutable, central 
force was nothing but commonplace gravity, known to exist 
on and near the earth. He verified his idea first in the case 
of the moon. He discovered that the same acceleration that 
controls the motion of an object near the earth also pre- 



NEWTON'S VERIFICATION 481 

vented the moon from moving away in a rectilinear path 
from the earth, and that its tangential velocity prevented 
it from falling to the earth. 

Assuming that the moon's orbit is circular, its acceleration 
towards the earth is (by Art. 70 (2)), 

a = ^ = ^YT = 0.0089 ft./sec.^, 

where R = 238,800 miles and T = 27.32 days. 
From the law of inverse squares : 

a _ r^ _ r^ _ 1 ^ 

'g~W~ (60.267r)2~3632' 

a 32.089 



3632 3632 



0.0088 ft./sec2., 



where 32.089 is the value of g on the earth at the equator, 
and R is 60.267 times r, the radius of the earth. 

As these results differ by only rooooth of a foot, the 
conclusion is that the centripetal force on the moon in its 
orbit is due to the earth's attraction, acting according to 
the law of inverse squares. 

Owing to an inaccurate value of the earth's radius which 
was in use at the time Newton first made the computation, 
the result then obtained seemed to show that the law of 
attraction was not that of inverse squares. 

Records show that Newton, although unshaken in his 
behef, laid aside his calculations; and it was not until 
thirteen years afterwards that, a new determination of the 
radius having been made, he repeated the investigation and 
found the verification sought for. 

Five years later (in 1684) he was induced to consider the 
whole subject of gravitation; and then he solved the supple- 
mentary problems in regard to the attraction of a sphere 
for an external particle, which established his theory — 
now known as Newton's Law of Universal Gravitation. 



INDEX 



(Numbers refer to pages) 



Acceleration, 19, 21 

angular, 92 

gravity, 90 

normal, 20, 89, 94 

resolution of, 86 

tangential, 19, 86, 94 

total, 19 
Agnesi, witch of, 132 
Algebraic functions, 9, 38 
Anti-derivatives, 263 
Anti-differential, 171 
Application of Taylor's theorem, 

424 
Applications, 99, 355, 456 

beams, 229 

prismoid formula, 284 
Approximate formulas, 138, 249 

relative rates and errors, 162 
Approximation formulas, 421 
Arc, differential of, 17 

infinitesimal, 37 

length of, 64 
Archimedes, 110, 142 
Area, any surface, 311 

as an integral, 223 

by double integration, 304, 308 

derivative of, 205 

finite or infinite, 208 

hyperbola, equilateral, 221 

positive or negative, 208 

under a curve, 206, 214 

under derived curves, 225 



Argument, 6 

Atmosphere, limit of, 378 
Attraction, 363 
Auxihary equation, 454 
theorems, 87 

Base, 6, 48, 56 

common, 54, 55 

Naperian or natural, 49, 50, 54 
Beams, 229 
Binomial theorem, 419 
Brachystochrone, 464 

Cable, 81 
Calculus, 1, 2, 17 

Differential, 3, 17, 100 

Integral, 3, 171 
Cardioid, 142, 217, 310, 311 
Catenary, 81, 140, 148, 245 

surface generated by, 293 

uniform strength, 251 

volume generated by, 294 
Cauchy's remainder, 416 
Center of curvature, 135, 142 

of gravity, 332, 333, 334, 335 
Central forces, 472 
Centrifugal force, 90, 378 

railway, 466 
Centripetal force, 90 
Centroid, 332 

Change of independent variable, 
149 



483 



484 



INDEX 



Change, rate of, 1, 2, 5 

uniform and non-uniform, 14 
Chrystal's Algebra, 421 
Circle, area of, 217, 218, 261, 
309 

curvature of, 135 

equation of, 5 

length of circumference, 237 
Circular functions, 64 

measure, 64, 70 

motion, 20, 88 

pendulum, 461 
Cissoid, length of, 237 

volume generated by, 295 
Coefficient, differential, 24 

of contraction, 469 

of velocity, 468 
Comparison test, 391 
Complementary function, 453 
Complete integral, 437 
Compound interest law, 9, 60, 

442 
Concavity, 122, 123 
Cone, 267, 270 
Conic section, 475 
Conoid, 282 
Constant, 5' 

absolute, 5, 6 

arbitrary, 5, 6 

of integration, 173, 301 
Continuity, 7 
Continuous functions, 7 

variables, 5, 7 
Convergence, tests for, 390, 393 
Convergent series, 388 
Cosine, differential of, 64, 65 

graph of, 119, 228 
Cubical parabola, 140, 237 
Curvature, 133 

center of, 135 

circle of, 135 

radius of, 135, 137, 140 



Curve, 8 

of cord with load uniform hori- 
zontally, 241 

of flexible cord, 245 

polar, 123 
Curvilinear motion, 460 
Curves, derived, 118, 225 

integral, 227 

lengths of, 236, 239 
Cusps, 114, 140, 146 
Cycloid, 105 

area of, 217 

equation of, 107 

evolute of, 147 

length of, 147, 237 

surface generated by, 293 

volume generated by, 294 
Cycloidal pendulum, 464 
Cylinder, 267, 279, 280, 288, 290, 

291 
Cylindrical coordinates, 324 

shce, 317 

surface, 318 

Damped vibrations, 73, 441, 455 
Decreasing function, 8 
Definite integral, 206, 213 
Deflection of beams, 141 
Density of air, 62 

mean, 327, 374 
Dependent variable, 6 
Derivative, 24, 25 

as a limit, 26 

as slope of curve, 29 

of area, 205 

partial, 153, 164 

successive, 84, 149 

total, 159 
Derived curves, 118, 225 

function, 24 
Differential Calculus, 3, 17 

coefficient, 24 



INDEX 



485 



Differential equations, 436, 441 

of sine in degrees, 69 
Differentials, 3, 15 

exact, 167 

inexact, 168 

partial, 152, 164 

successive, 84 

total, 156 
Differentiation, 2, 25 

of series, 394 

rules of, 64 
Discharge from an orifice, 469 
Discontinuous function, 7 

variables, 5 
Divergency of series, 391 
Double integration, 304, 308, 324 

E, modulus of elasticity, 230, 254 
e, Napierian base, 6, 50, 418 
Eccentricity, 475 
Elastic curve, 230 
Elasticity, modulus of, 230, 254 
Elementary principles, 178 
Elements, 261, 331 
Ellipse, area of, 218 

length of quadrant, 238, 398 
Ellipsoid, 292, 294 
Elliptic integral, 400, 462, 463 
Empirical equations, 10 
Energy integral, 453 

kinetic, 92 
Equation of evolute, 144 

of involute of circle, 146 

of involute of cycloid, 146 

of normal, 100 

of tangent, 99 
Equations, differential, 436 

exact differential, 444 

homogeneous, 446 

linear, 448 

of first order, 449 

of order above first, 449 



Equilateral hyperbola, 140, 147, 

221 
Error term, 421 
Errors, percentage, 163 
Euler's series, 397 
Evaluation of definite integrals, 
213 
of derivatives of implicit func- 
tions, 435 
of indeterminate forms, 428 
Evolute, 143, 144, 256 
Evolution, 3 
Exact differential equations, 167, 

170, 444 
Examples, illustrative, 31, 76, 160, 

174, 412 
Expansion by Maclaurin's and by 
Taylor's Theorems, 411 
of cosh x/a and sinh x/a, 248 
of fimctions in series, 408 
Explicit function, 6 
Exponential function, 9, 47 
Extended law of the mean, 405 

Factor, integrating, 445 
FalUng bodies, 21, 177, 297 
First moment, 331 
Flexion, 21, 133 
Force, central, 472 

centrifugal, 90, 378 

centripetal, 90 

concentrated, 355 

definition of, 92 

distributed, 355 

variable, 472 
Forms, indeterminate, 425, 428 

standard, 180, 181, 183, 203, 403 
Formula, prismoid, 283 

projectile, Helie's, 384 
Formulas, 38, 47, 64 

approximation, 421 

reduction, 194, 198 



486 



INDEX 



Frustum, surface and volume of 

any, 270 
Function, 6 

algebraic, 9, 38 

continuous, 7 

discontinuous, 7 

exponential, 9, 47 

hyperbolic, 9, 80 

inverse, 76, 82 

logarithmic, 9, 47 

of a function, 27 

power, 9, 10 

transcendental, 9 

trigonometric, 9, 64 

several variables, 152 
Functional relation, 7 
Fundamental conception, 30 

condition or test, 112 

rule for applying test, 115 

theorem, 261 

g, acceleration of gravity, 90 
Gas, formula for, 160 
Gauge, self-registering, 97 
Geometric meaning of integral, 

204 
Geometric progression, 9, 10 

series, 385 
Grade, 23 

Graphical illustration, 113 
Graphs, 8, 80, 118 

cosine, 119, 228 

cycloid, 105 

sine, 71, 119, 228 

versine, 228 
Gravitation, law of, 363, 371, 481 

unit of mass, 376 
Gravity, acceleration of, 90, 91 

center of, 332, 333, 334 
Gregory's series, 397 
Guldin's theorems, 336 
Gyration, radius of, 345 



Harmonic law, 9 

motion, 94, 442, 457 

series, 389, 392 
Helie's formula, 384 
Homogeneous equations, 446 
Hyperbola, equilateral, 140, 221 
Hyperbolic functions, 9, 80 

logarithms, 216 
Hypocycloid, 140, 148 

Ideal quantity, 2 
Ideas, 2 
Illustrations, 21 

typical, 118 
Illustrative examples, 31, 76, 160, 

174, 412 
Implicit function, 7 
Increasing function, 8 
Increment, 12 
Indefinite constant, 173, 301 

integral, 173, 301 
Independent variable, 6, 149 
Indeterminate forms, 425 
Inertia, 91 

moment of, 230, 345, 346, 349, 
350, 352 

product of, 348 
Inexact differential, 168 
Infinite series, 385 
Infinitesimal, 30, 36, 37 

arc and chord, 37 
Infinity, 36 
Inflexion, point of, 114, 120, 132, 

140 
Integral, complete, 437,- 453 

Calculus, 3, 171 

definite, 206 

energy, 453 

from an area, 219 

general, 227 

indefinite, 173, 301 

multiple, 297 



INDEX 



487 



Integral, particular, 175, 453 
Integrals, elliptic, 400, 462, 463 
Integrand, 171 
Integrating factor, 445 
Integration, 2, 17, 170, 171 

double, 304, 308, 324 

constant of, 173, 301 

parts, 194 

series, 394 

successive, 296, 303 

triple, 317, 321 
Intensity, 355 
Interchange of limits, 210 
^of order of differentiation, 165 

of order of integration, 299, 301, 
306 
Interest, compound, law of, 9, 60 
Inverse functions, 9 

hyperbolic functions, 81 

of differentiation, 171 

trigonometric functions, 76 
Involute, 143 

of the catenary, 248 
Involution, 3 

Jet, liquid, path of, 467 

Kepler's laws, 477 
Kinetic energy, 92 

Lagrange's remainder, 407 
Law, compound interest, 9, 60, 
442 

extended, of the mean, 405 

of organic growth, 9, 62, 442 

of the mean, 401, 402, 403 

of motion, 90, 359, 365 

parabolic, 9 

planetary motion, 477 

universal gravitation, 363, 371, 
481 
Lemniscate, 110, 142, 217, 326 



Lengths of curves, 236, 239 
Limit of a sum, 259 

of height of atmosphere, 378 

of infinitesimal arc, 37 
Limits, 25, 26, 30, 206, 208, 210 
Linear equations, 448, 453 
Liquid jet, 467 

pressure, 356 
Lituus, 124 
Locus, 8 
Logarithmic differentiation, 56 

curve, 140 

functions, 47 

series, 395 

spiral, 142 
Logarithms, 54, 55 

common, 55 

hyperbolic, 216 

natural, 55 

Machin's series, 397 
Maclaurin's theorem, 407 

series, 408 
Mass, 91, 327 

unit of, 376 
Maxima and minima, 111 

application of Taylor's theorem, 
to, 424 

problems, 127 
Maximum and minimum. 111 
Mean density, 327, 374 

law of the, 401, 402, 403 

value of a function, 211 
Method of the Calculus, 430 

of limits, 30 
Modulus, 54 

of elasticity, 230, 254 
Moment, 331, 345, 355 

of inertia, 230, 350 

least moment of inertia, 349 

of inertia for parallel axes, 346 

principal moment of inertia, 348 



488 



INDEX 



Moments of area, 333, 352 

first, 331 

of line, 334 

second, 344 

of volume, 333 
Momentum, 92 
Motion, circular, 88 

curvilinear, 460 

in resisting medium, 379, 383, 
459 

planetary, 477 

rectilinear, 456 

second law of, 90 

simple harmonic, 94, 442, 457 

third law of, 90, 359, 365 

•Napierian or natural base„ 49, 50, 

54 
. logarithms, 55, 70, 216 
Newton, 4, 90, 359, 363, 371, 

480 
Normal acceleration, 20, 89, 94 
Normal, 99 

polar, 109 
Notation for functions, 6 
Number e, 6, 50, 418 

TT, 6, 50, 398, 418 

Orbit, 472, 475 

Order above the first, 449 

first, 449 

of a differential equation, 436 
Ordinary differential equation, 

436, 444 
Organic growth, 9, 62, 442 
Orifice, discharge from, 469 
Oscillating series, 385, 397 

^, 6, 50, 398, 418 
Pappus, theorem of, 336 
Parabolic cable or cord, 241 
law, 9 



Partial derivations, 153, 164 

differentials, 152, 164 
Particular integral, 175, 453 

values, 6 
Parts, integration by, 194 
Path of a projectile, 381 

of a liquid jet, 467 
Pendulum, simple circular, 461 

cycloidal, 464 
Percentage rate, 56 

error, 163 
Period, 96 
Periodic time, 477 
Plane, tangent, 154 
Plane areas, 304, 308 
Planetary motion, 477, 478 
Points of inflexion, 114, 120, 132, 

140 
Polar curve, 8, 123 

moment of inertia, 345 

subtangent, subnormal, 108 
Power form, 183 

formula, 44 

function, 9, 10 

series, 386, 393 
Pressure of air, 62 

hquid, 356 
Primitive of differential equation, 

437 
Principles, 371, 480 
Prismoid formula, 283, 284 
Probability integral, 418 
Problems, maxima and minima, 

127 
Process, summation, 261 
Product of inertia, 348 
Progression, 9, 10 
Projectile, path of, 381 

Quadrature of curves, 443 
Quotient, differential of, 39, 42 
limit of, 26, 35 



INDEX 



489 



Radian, 64, 70 
Radium, 63 

Radius of curvature, 135, 137, 
140 

of gyration, 345 
Railway, centrifugal, 466 
Range formula, Helie's, 384 
Rate, 18, 19, 21 

of change, 1, 2 

percentage, 56 

relative, 56, 162 
Rectification of curves, 237 
Rectilinear motion, 456 
Relative error, 59, 162 

rate, 56, 162 
Remainder, Cauchy's, 416 

Lagrange's, 407 
Remarks, 27, 70 
Replacement theorem, 37 
Representation of functional re- 
lation, 7 

volume by area, 269 
Resisting medium, 379, 383, 459 
Rotation, 92 

Secant, 29, 65, 68 
Second derivative, 84, 86 

moments, 344 
Semi-cubical parabola, 145, 237 
Separable variable, 446 
Separated differential equation, 

444 
Separation into parts, 210 
Series, infinite, 385 

absolutely convergent, 388 

convergent, 385, 393 

divergent, 385 

Euler's, 397 

for e, 50 

Gregory's, 397 

geometric, 385 

harmonic, 389, 392 



Series, integration and differen- 
tiation of, 394 

logarithmic, 395 

Machin's, 397 

Maclaurin's, 408 

Newton's, 397 

oscillating, 385, 397 

power, 386, 393 

Taylor's, 409 
Shear, 229 
Shooting point, 114 
Significance of area, 223 
Simple circular pendulum, 461 

harmonic motion, 94, 442, 457 
Sine curve, 72 

differential of, 64, 65 

graph of, 71, 119, 228 

ratio to arc, 66, 67 
Slope, 18, 23, 99 

rate of change of, 21 
Solids of revolution, 288 

by double integration, 321 
Solution of a differential equation, 
436 

of s = a sinh x/a, 250 
Speed, 1, 18, 19 
Sphere, 268, 274, 285, 313, 323 
Spherical shell, 368, 371 
Spheroid, 294 
Spiral of Archimedes, 110, 142 

logarithmic, 110, 142 
Standard forms, 180, 182, 203 
Statical moment, 345, 359 
Stiffness of beams, 128 
Strength of beams, 127 
Subnormal, 100, 108 
Subtangent, 100, 108 
Successive derivatives, 84, 149 

differentials, 89 

differentiation, 84 

integration, 296, 300 
Summation process, 261 



490 



INDEX 



Summation process, approximate 

and exact, 262 
Surface of any frustum, 270 

of revolution, 288 
Suspension bridge, 244 

Tangent, equation of, 99 

plane, 154 

polar, 109 
Tangential acceleration, 19, 86, 94 
Taylor's theorem, 405, 406, 407 

series, 409 
Test, comparison, .391 

for convergence, 390 

ratio, 391 
Theorem, replacement, 37 

binomial, 419 

finite differences, 402 

Maclaurin's, 407 

Taylor's, 405, 406, 407 
Theorems, auxiliary, 124 

of limits, 26 

of mean value, 401, 402 
Theorems of Pappus and Guldin, 

336 
Tide gauge, 97 
Time, periodic, 477 

rate of change, 19 
Total derivative, 159 

differential, 156 
Tractrix, 254 
Transcendental functions, 9 

numbers, 6, 50 
Trapezoid, 343 

Trigonometric functions, 9, 64 
Triple integration, 317, 321 
Typical illustrations, 118 



Uniform change, 14 

speed or velocity, 20 
Unit of force, 91 

of mass, 91, 376 
Universal gravitation, 363, 371, 

481 
Use of standard formulas, 1^ 

Value, mean, 211 
Variable, 5 

continuous, 5 

dependent, 6 

discontinuous, 5 

independent, 6, 149 

separable, 446 
Vector quantity, 21, 355 
Velocity, 18 

angular, 92 

average, 22, 23 

constant, 20 

instantaneous, 1 

on a curve, 94 

tangential, 94 

uniform, 20 
Verification, Newton's, 480 
Vibrations, damped, 73 
Volume by an area, 268 

of frustum, 270 
Volumes, 267 

by double integration, 324 

by triple integration, 317, 321 

Water pressure, 357 
Wave curve, 72 
Wetted perimeter, 129 
Witch of Agnesi, 132 
Work, 167 



^ _ ^ 

D. VAN NOSTRAND COMPANY 
25 PARK PLACE 

NEW YORK 



SHORT-TITLE CATALOG 



OF 



PnNications i Titiportations 

OF 

SCIENTIFIC AND ENGINEERING 
BOOKS 




This list includes the technical publications of the following 
English publishers: 

SCOTT, GREENWOOD & CO. JAMES MUNRO & CO., Ltd. 

CONSTABLE & COMPANY, Ltd. TECHNICAL PUBLISHING CO. 

ELECTRICIAN PRINTING & PUBLISHING CO., 

for whom D. Van Nostrand Company are American agents 

Descriptive Circulars sent on request. 



October, I9i8 

SHORT=TITLE CATALOG 

OP THE 

Publications and Importations 

OF 

D. VAN NOSTRAND COMPANY 

25 PARK PLACE 

Prices marked with an asterisk {*) are NET 
All bindings are in cloth unless otherwise noted 



Abbott, A. V. The Electrical Transmission of Energy 8vo, *$5 oo 

— — A Treatise on Fuel. (Science Series No. 9.) . i6mo, o 50 

Testing Machines. (Science Series No. 74.) : . i6mo, o 50 

Abraham, H. Asphalt and Allied Substances 8vo, 5 00 

Adam, P. Practical Bookbinding. Trans by T. E. Maw.iamo, *3 00 

Adams, H. Theory amd Practice in Designing 8vo, *2 50 

Adams, H. C. Sewage of Seacoast Towns. 8vo, ^2 50 

Adams, J. W. Sewers and Drains for Populous Districts — 8vo, 2 50 

Adler, A. A. Theory of Engineering Drawing Svo, *2 00 

Principles of Parallel Projecting-Line Drawing Svo, *i 00 

Aikman, C. M. Manures and the Principles of Manuring. . . Svo, 2 50 

Aitken, W. Manual of the Telephone Svo, *8 00 

d'Albe, E. E. F. Contemporary Chemistry i2mo, *i 25 

Alexander, J. H. Elementary Electrical Engineering. .. .izmo, 2 50 



D. VAN NOSTRAND COMPANY S SHORT-TITLE CATALOG 3 

Allan, W. Strength of Beams under Transverse Loads. 

(Science Series No. 19.) i6mo, o 50 

Allan, W. Theory of Arches. (Science Series No. 11.). . i6mo, 

Allen, H. Modern Power Gas Producer Practice and Applica- 
tions i2mo, *2 50 

Anderson, J. W. Prospector's Handbook i2mo, i 50 

Andes, L. Vegetable Fats and Oils 8vo, *7 25 

Animal Fats and Oils. Trans, by C. Salter 8vo, *5 00 

Drying Oils, Boiled Oil, and Solid and Liquid Driers. .8vo, *7 50 

Iron Corrosion, Anti-fouling and Anti-corrosive Paints. 

Trans, by C. Salter Svo, *6 00 

Oil Colors and Printers' Ink. Trans, by A. Morris and 

H. Robson Svo, *4 00 

Treatment of Paper for Special Purposes. Trans, by C. 

Salter lamo, *3 00 

Andrews, E. S. Reinforced Concrete Construction i2mo, *2 00 

Theory and Design of Structures Svo, 3 50 

Further Problems in the Theory and Design of Struc- 
tures Svo, 2 50 

The Strengtii of Materials Svo, *4 00 

Andrews, E. S., and Heywood, H. B. The Calculus for 

Engineers lamo, 2 00 

Annual Reports on the Progress of Chemistry. Twelve Vol- 
umes now ready. Vol. I, 1904, to Vol. XII, 1914, Svo, 
each *2 00 

Argand, M. Imaginary Quantities. Translated from the French 

by A. S. Hardy. (Science Series No. 52.) i6mo, o 50 

Armstrong, R., and Idell, F. E. Chimneys for Furnaces and 

Steam Boilers. (Science Series No. i.) i6mo, o 50 

Arnold, E. Armature Windings of Direct Current Dynamos. 

Trans, by F. B. DeGress Svo, *2 00 

Asch, W., and Asch, D. The Silicates in Chemistry and 

Commerce Svo, *6 00 

Ashe, S. W., and Keiley, J. D. Electric Railways. Theoreti- 
cally and Practically Treated. Vol. I. Rolling Stock 

i2mo, *2 so 



4 D. VAN NOSTRAND COMPANY S SHORT-TITLE CATALOG 

Aghe, S. W. Electric Railways. Vol. II. Engineering Pre- 

liminarie and Direct Current Sub-Stations i2mo, *2 50 

-Electricity: Experimentally and Practically Applied. 

i2mo, *2 00 

Ashley, R. H. Chemical Calculations i2mo, *2 00 

Atkins, W. Common Battery Telephony Simplified. .. .i2mo, *i 25 

Atkinson, A. A. Electrical and Magnetic Calculations .. 8vo, *i 50 
Atkinson, J. J. Friction of Air in Mines. (Science Series 

No. 14. ) i6mo, o 50 

Atkinson, J J., and Williams, E. H., Jr. Gases Met with in 

Coal Mines. (Science Series No. 13.) i6mo, o 50 

Atkinson, P. The Elements of Electric Lighting i2mo, i 00 

TheElementsof Dynamic Electricity and Magnetism. 1 2mo, 2 00 

Atkinson, P. Power Transmitted by Electricity i2mo, 2 00 

Auchincloss, W. S. Link and Valve Motions Simplified. .. .8vo, *i 50 

Austin, E. Single Phase Electric Railways .4to, *5 00 

Austin and Cohn. Pocketbook of Radiotelegraphy {In Press.) 

Ayrton, H. The Electric Arc 8vo, *5 50 

Bacon, F. W Treatise on the Richards Steam-Engine Indica- 
tor , i2mo, I 00 

Bailey, R. D. The Brewers' Analyst 8vo, *5 00 

Baker, A. L. Quaternions 8vo, *i 25 

Thick-Lens Optics i2mo, *i 50 

Baker, Benj. Pressure of Earthwork. (Science Series No. 56.) 

i6mo. 
Baker, G. S. Ship Form, Resistance and Screw Propulsion, 

Svo, *4 50 

Baker, I. Levelling. (Science Series No. 91.) i6mo, o 50 

Baker, M. N. Potable Water. (Science Series No. 61) . i6mo, o 50 
Sewerage and Sewage Purification. (Science Series No. 18.) 

i6mo, o 50 
Baker, T. T. Telegraphic Transmission of Photographs. 

i2mo, *i 25 
Bale, G. R. Modern Iron Foundry Practice. Two Volumes. 

i2mo. 

Vol. I. Foundry Equipment, Material Used *3 00 

Vol. II. Machine Moulding and Moulding Machines *i 50 



D. VAN NOSTRAND COMPANY'S SHORT-TITLE CATALOG 5 

Ball, J. W. Concrete Structures in Railways 8vo, *2 50 

Ball R. S. Popular Guide to the Heavens 8vo, 5 00 

Natural Sources of Power. (Westminster Series) 8vo, *2 00 

Ball, W. V. Law Affecting Engineers 8vo, *3 50 

Bankson,, Lloyd. Slide Valve Diagrams. (Science Series No. 

108.) i6mo, 50 

Barham, G. B. Development of the Incandescent Electric 

Lamp 8vo; *3 00 

Barker, A. F. Textiles and Their Manufacture. (Westminster 

Series) 8vo, 2 00 

Barker, A. F., and Midgley, E. Analysis of Woven Fabrics, 

Bvo, 4 25 

Barker, A. H. Graphic Methods of Engine Design. .. .i2mo, *2 00 

Heating and Ventilation 4to, *8 00 

Barnard, J. H. The Naval Militiaman's Guide. .i6mo, leather, i 00 
Barnard, Major J. G. Rotary Motion. (Science Series No. 90.) 

i6mo, 50 

Barnes, J. B. Elements of Military Sketching i6mo, *o 75 

Barrus, G. H. Engine Tests 8vo, *4 00 

Barwise, S. The Purification of Sewage i2mo, 3 50 

Baterden, J. R. Timber. (Westminster Series) 8vo, *2 00 

Bates, E. L., and Charlesworth, F. Practical Mathematics and 
Geometry i2mo, 

Part I. Preliminary and Elementary Course *i 50 

Part II. Advanced Course *i 50 

— — Practical Mathematics i2mo, *2 00 

Practical Geometry and Graphics lamo, *2 00 

Batey, J. The Science of Works Management i2mo, *i 50 

Steam Boilers and Combustion lamo, *i 50 

Bayonet Training Manual i6mo, o 30 

Beadle, C. Chapters on Pa permaking. Five Volumes . 1 2mo, each, *2 00 

Beaumont, R. Color in Woven Design Svo, *6, 00 

Finishing of Textile Fabrics Svo, *7 50 

Standard Cloths Svo, *7 50 

Beaumont, W. W. Steam-Engine Indicator Svo, 2 50 



6 D. VAN NOSTRAND COMPANY'S SHORT-TITLE CATALOG 

Bechhold, H. Colloids in Biology and Medicine. Trans, by J. G. 

Bullowa {In Press.) 

Beckwith, A. Pottery 8 vo, paper, o 60 

Bedell, F., and Pierce, C. A. Direct and Alternating Current 

Manual » . 8vo, *2 00 

Beech, F. Dyeing of Cotton Fabrics 8vo, *7 50 

Dyeing of Woolen Fabrics 8vo, *4 25 

Beggs, G. E. Stresses in Railway Girders and Bridges . . . .{In Press.) 

Begtrup, J. The Slide Valve Svo, *2 00 

Bender, C. E. Continuous Bridges. (Science Series No. 26,) 

i6mo, o 50 

Proportions of Pins used in Bridges. (Science Series No. 4.) 

i6mo, o 50 

Bengough, G. D. Brass. (Metallurgy Series) {In Press.) 

Bennett, H. G. The Manufacture of Leather Svo, *5 00 

Bernthsen, A. A Text-book of Organic Chemistry. Trans, by 

G. M'Gowan i2mo, ''3 00 

Bersch, J. Manufacture of Mineral and Lake Pigments. Trans. 

by A. C. Wright Svo, *5 00 

Bertin, L. E. Marine Boilers. Trans, by L. S. Robertson .. Svo, 5 00 

Bevexidge, J. Papermaker's Pocket Book i2mo, "^4 00 

Binnie, Sir A. Rainfall Reservoirs and Water Supply . . 8vo, *3 00 

Binns, C. F. Manual of Practical Potting Svo, *io 00 

The Potter's Craft i2mo, *2 00 

Birchmore, W. H. Interpretation of Gas Analysis i2mo, *i 25 

Blaine, R. G. The Calculus and Its Applications i2mo, *i 75 

Blake, W. H. Brewer's Vade Mecum Svo, *4 00 

Blasdale, W. C. Quantitative Chemical Analysis. (Van 

Nostrand's Textbooks) i2mo, *2 50 

Bligh, W. G. The Practical Design of Irrigation Works. .Svo, 
Bloch, L. Science of Illumination. (Trans, by W. C. 

Clinton) Svo, *2 50 

Blok, A. Illumination and Artificial Lighting i2mo, *2 25 

Blucher, H. Modern Industrial Chemistry. Trans, bj^ J. P. 

Millington Svo, *7 50 

Blyth, A. W. Foods: Their Composition and Analysis. ..8vc, 7 50 

• Poisons: Their Effects and Detection Svo, 8 50 



D. VAN NOSTRAND COMPANY S SHORT-TITLE CATALOG- 7 

Bockmann, F. Celluloid i2mo, *3 oo 

Bodmer, G. R. Hydraulic Motors and Turbines lamo, 5 00 

Boileau, J. T. Traverse Tables 8vo, 5 00 

Bonney, G. E. The Electro-plater's Handbook lamo, i 50 

Booth, N. Guide to Ring-Spinning Frame lamo, '^2 00 

Booth, W. H. Water Softening and Treatment 8vo, *2 50 

Superheaters and Superheating and their Control. . .8vo, *i 50 

Bottcher, A. Cranes: Their Construction, Mechanical Equip- 
ment and Working. Trans, by A. Tolhausen. .. .4to, *io 00 
Bottler, M. Modern Bleaching Agents. Trans, by C. Salter. 

i2mo, *3 00 

Bottone, S. R. Magnetos for Automobilists i2mo, *i 00 

Boulton, S. B. Preservation of Timber. (Science Series No. 

82.) i6mo, o 50 

Bourcart, E. Insecticides, Fungicides and Weedkillers . . . 8vo, *7 50 
Bourgougnon, A. Physical Problems. (Science Series No. 113.) 

i6mo, o 50 
Bourry, E, Treatise on Ceramic Industries. Trans, by 

A. B. Searle 8vo, *7 25 

Bowie, A. J., Jr. A Practical Treatise on Hydraulic Mining . 8vo, 5 00 
Bowles, 0. Tables of Common Rocks. (Science Series, 

No. 125) i6mo, o 50 

Bowser, E. A. Elementary Treatise on Analytic Geometry. i2mo, i 75 

Elementary Treatise on the Differential and Integral 

Calculus i2mo, 2 25 

Bowser, E. A. Elementary Treatise on Analytic Mechanics, 

i2mo, 3 00 

Elementary Treatise on Hydro-mechanics i2mo, 2 50 

A Treatise on Roofs and Bridges i2mo, *2 25 

Boycott, G. W. M. Compressed Air Work and Diving .... 8vo, *4 25 
Bradford, G., 2nd. Whys and Wherefores of Navigation, 

i2mo, 2 00 

Sea Terms and Phrases i2mo, fabrikoid {In Press.) 

Bragg, E. M. Marine Engine Design i2mo, *2 00 

Design of Marine Engines and Auxiliaries *3 00 

Brainard, F. R. The Sextant. (Science Series No. ioi.).i6mo, 

Brassey's Naval Annual for 19 15. War Edition 8vo, 4 00 

Briggs, R., and Wolff, A. R. Steam-Heating. (Science Series 

No. 67.) i6mo, 050 



8 D. VAN NOSTRAND COMPANY^'s SHORT-TITLE CATALOG 

Bright, C. The Life Story of Sir Charles Tilson Bright. .8vo, ^4 50 

Telegraphy, Aeronautics and War 8vo, 6 00 

Brislee, T. J. Introduction to the Study of Fuel. (Outlines 

of Industrial Chemistry.) 8vo, *3 00 

Broadfoot, S. K. Motors Secondary Batteries. (Installation 

Manuals Series.) i2mo, *o 75 

Broughton, H. H. Electric Cranes and Hoists. 

Brown, G, Healthy Foundations. (Science Series No. 80.). i6mo, o 50 

Brown, H. Irrigation 8vo, *5 00 

Brown, H, Rubber 8vo, *2 00 

Brown, W. A. Portland Cement Industry 8vo, 3 00 

Brown, Wm. N. The Art of Enamelling on Metal. .. .lamo, *2 25 

Handbook on Japanning i2mo, '^2 50 

— ■ — House Decorating and Painting i2mo, *2 25 

History of Decorative Art i2mo, *i 25 

Dipping, Burnishing, Lacquering and Bronzing Brass 

Ware i2mo, *2 00 

Workshop Wrinkles 8vo, *i 75 

Browne, C. L. Fitting and Erecting of Engines 8vo, *i 50 

Browne, R. E. Water Meters. (Science Series No. 8i.).i6mo, o 50 
Bruce, E. M. Detection of the Common Food Adulterants, 

i2mo, I 25 
Brunner, R. Manufacture of Lubricants, Shoe Polishes and 

-Leather Dressings. Trans, by C. Salter 8vo, *4 50 

Buel, R. H. Safety Valves. (Science Series No. 21.) . . . i6mo, o 50 

Bunkley, J. W. Military and Naval Recognition Book. .i2mo, i 00 
Burley, G. W. Lathes, Their Construction and Operation, 

i2mo, 2 25 

Machine and Fitting Shop Practice i2mo, 2 50 

Testing of Machine Tools i2mo, 2 50 

Burnside, W. Bridge Foundations i2mo, *i 50 

Burstall, F. W. Energy Diagram for Gas. With text . . . 8vo, i 50 

Diagram sold separately *i 00 

Burt, W. A. Key to the Solar Compass i6mo, leather, 2 50 

Buskett, E. W. Fire Assaying lamo, *i 25 

Butler, H. J. Motor Bodies and Chasis 8vo, *-3 00 

Byers, H. G., and Knight, H. G. Notes on Qualitative 

Analysis 8vo, *i 50 

Cain, W. Brief Course in the Calculus i2mo, *i 75 

Elastic Arches. (Science Series No. 48.) i6mo, o 50 



D. VAN NOSTRAND COMPANY S SHORT-TITLE CATALOG 9 

Maximum Stresses. (Science Series No. 38.) i6mo, 50 

^Practical Dsigning Retaining of Walls. (Science Series 

No. 3.) •. i6mo, o 50 

Theory oi Steel-concrete Arches and of Vaulted Struc- 
tures. (Science Series, No. 42) i6mo, o 75 

Theory of Voussoir Arches. (Science Series No. 12.) 

i6mo, o 50 

Symbolic Algebra. (Science Series No. 73.) i6mo, o 50 

Calvert, G. T. The Manufacture of Sulphate of Ammonia 

and Crude Ammonia i2mo, 4 00 

Carpenter, F. D. Geographical Surveying. (Science Series 

No. 37.) i5mo, 

Carpenter, R. C, and Diederichs, H. Internal-Combustion 

Engines 8vo, *5 00 

Carter, H. A. Ramie (Rhea), China Grass i2mo, *3 00 

Carter, H. R. Modem Flax, Hemp, and Jute Spinning. .Svo, *4 50 

Bleaching, Dyeing and Finishing of Fabrics Svo, *i 25 

Cary, E. R. Solution of Railroad Problems With the Use of 

the Slide Rule i6mo, *i 00 

Casler, M. D. Simplified Reinforced Concrete Mathematics, 

i2mo, 
Cathcart, W. L. Machine Design. Part I. Fastenings. . .Svo, 
Cathcart, W. L., and Chaffee, J. I. Elements of Graphic 

Statics . Svo, 

Short Course in Graphics i2mo, 

Caven, R. M., and Lander, G. D. Systematic Inorganic Chem- 
istry i2mo, 

Chalkley, A. P. Diesel Engines Svo, 

Chalmers, T, W, The Production and Treatment of Veg- 
etable Oils 4to, 

Chambers' Mathematical Tables Svo, 

Chambers, G. F. Astronomy i6mo, 

Chappel, E. Five Figure Mathematical Tables Svo, 

Charnock. Mechanical Technology Svo, 

Charpentier, P. Timber Svo, 

Chatley, H. Principles and Designs of Aeroplanes. (Science 

Series".) i6mo, 

How to Use Water Power. , . . . i2mo, 

— — Gyrostatic Balancing Svo, *i 25 



*I 


GO 


*3 


00 


*3 


00 


I 


50 


*2 


00 


*4 


00 


7 


50 


I 


75 


*i 


50 


*2 


00 


^•'3 


00 


*7 


25 





50 


-,-j 


50 



10 D. VAN N0S1\-.AND COMPANY S SHORT-TITLE CATALOG 

Child, C. D. Electric Arcs 8vo, *2 oo 

Christian, M. Disinfection and Disinfectants. Trans, by 

Chas. Salter i2mo, 3 00 

Christie, W. W. Boiler-waters, Scale, Corrosion, Foaming, 

8vo, *3 00 
Chimney Design and Theory 8vo, *3 00 

Furnace Draft. (Science Series, No. 123) i6mo, o 50 

Water, Its Purification and Use in the Industries. . . .Svo, *2 00 

Church's Laboratory Guide. Rewritten by Edward Kinch.Svo, *i 50 

Clapham, J. H. Woolen and Worsted Industries Svo, 2 00 

Clapperton, G. Practical Papermaking Svo, 250 

Clark, A. G. Motor Car Engineering. 

Vol. I. Construction *3 00 

Vol. II. Design (In Press.) 

Clark, C. H. Marine Gas Engines. New Edition 2 00 

Clark, J. M. New System of Laying Out Railway Turnouts, 

t i2mo, I 00 
Clarke, J. W., and Scott, W. Plumbing Practice. 

Vol. I. Lead Working and Plumbers* Materials. .Svo, *4 00 

Vol. II. Sanitary Plumbing and Fittings (In Press.) 

Vol. III. Practical Lead Working on Roofs (In Press.) 

Clarkson, R. B. Elementary Electrical Engineering. 

(In Press.) 
Clausen-Thue, W. ABC Universal Commercial Telegraphic 

Code. Sixth Edition. (In Press.) 

Clerk, D., and Idell, F. E. Theory of the Gas Engine. 

(Science Series No. 62.) i6mo, o 50 

Clevenger, S. R. Treatise on the Method of Government 

Surveying i6mo, mor., 2 50 

Clouth, F. Rubber, Gutta-Percha, and Balata Svo, *6 00 

Cochran, J. Treatise on Cement Specifications Svo, *i 00 

Concrete and Reinforced Concrete Specifications. .. .Svo, *2 50 

Cochran, J. Inspection of Concrete Construction Svo, *4 00 

Cocking, W.C. Calculations for Steel-Frame Structures. lamo, 3 00 
Coffin, J. H. C. Navigation and Nautical Astronomy. . i2mo, *3 50 
Colburn, Z., and Thurston, R. H. Steam Boiler Explosions. 

(Science Series No, 2,) , i6nio, o 50 



D. VAN NOSTRAND COMPANY^S SHORT-TITLE CATALOG II 

Cole, R. S. Treatise on Photographic Optics ..i2mo, i 50 

Coles-Finch, W. Water, Its Origin and Use 8vo, *5 00 

Collins, C. D. Drafting Room Methods, Standards and 

Forms 8vo, 2 00 

Collins, J. E. Useful Alloys and Memoranda for Goldsmiths, 

Jewelers i6mo, o 50 

Collis, A. G. High and Low Tension Switch-Gear Design . 8vo, *3 50 

Switchgear. (Installation Manuals Series.) i2mo, o 50 

Comstock, D. F., and Troland, L. T. The Nature of Matter 

and Electricity. . . ' i2mo, 2 00 

Coombs, H. A. Gear Teeth. (Science Series No. 120). . . i6mo, o 50 

Cooper, W. R. Primary Batteries 8vo, *4 00 

Copperthwaite, W. C. Tunnel Shields 4to, *9 00 

Corfield, W. H. Dwelling Houses. (Science Series No. 50.) i6mo, o 50 

^ Water and Water-Supply. (Science Series No. 17.). . i6mo, 050 

Cornwall, H. B. Manual of Blow-pipe Analysis 8vo, *2 50 

Cowee, G. A. Practical Safety Methods and Devices. .. .Svo, *3 00 

Cowell, W. B. Pure Air, Ozone, and Water i2mo, *3 00 

Craig, J. W., and Woodward, W. P. Questions and Answers 

about Electrical Apparatus i2mo, leather, i 50 

Craig, T. Motion of a Solid in a Fuel. (Science Series No. 49.) 

i6mo, o 50 

Wave and Vortex Motion. (Science Series No. 43.) . i6mo, o 50 

Cramp, W. Continuous Current Machine Design Svo, *2 50 

Creedy, F. Single-Phase Commutator Motors Svo, *2 00 

Crehore, A. C. Mystery of Matter and Energy i2mo, i 00 

Crocker, F. B. Electric Lighting. Two Volumes. Svo. 

Vol. I. The Generating Plant 3 00 

Vol. II. Distributing Systems and Lamps 

Crocker, F B., and Arendt, M. Electric Motors ....Svo, *2 50 

and Wheeler, S. S. The Management of Electrical Ma- 
chinery i2mo, *i 00 

Cross, C. F., Bevan, E. J., and Sindall, R. W. Wood Pulp and 

Its Applications. (Westminster Series.) Svo, *2 00 

Crosskey, L. R. Elementary Perspective Svo, 1 25 

Crosskey, L. R., and Thaw, J. Advanced Perspective . . Svo, i 50 

Culley, J. L. Theory of Arches. (Science Series No. S7.). i6rao, o 50 
Cushing, H. C, Jr., and Harrison, N. Central Station Man- 
agement *2 00 

Dadourian, H. M. Analytical Mechanics Svo. *3 00 



12 D. VAN NOSTRAND COMPANY'S SHORT-TITLE CATALOG 

Dana, R. T. Handbook of Construction Plant. .i2mo, leather, *5 oo 

— —Handbook of Construction Efficiency {In Pres.s.) 

Danby, A. Natural Rock Asphalts and Bitumens 8vo, *2 50 

Davenport, C. The Book. (Westminster Series.) 8vo, *2 00 

Davey, N. The Gas Turbine 8vo, *4 00 

Da vies, F H. Electric Power and Traction 8vo, *2 00 

Foundations and Machinery Fixing. (Installation Manuals 

Series) i6mo, *i 00 

Deerr, N. Sugar Cane 8vo, 9 00 

Deite, C. Manual of Soapmaking. Trans, by S. T. King . . 4to, 
De la Coux, H. The Industrial Uses of Water. Trans, by A. 

Morris Svo, *6 00 

Del Mar, W. A. Electric Power Conductors Svo, *2 00 

Denny, G. A. Deep-Level Mipes of the Rand 4to, *io 00 

Diamond Drilling for Gold *5 00 

De Roos, J. D. C. Linkages. (Science Series No. 47.). . . i6mo, o 50 

Derr, W. L. Block Signal Operation Oblong i2mo, *i 50 

Maintenance of Way Engineering {In Preparation.) 

Desaint, A. Three Hundred Shades and How to Mix Them. 

Svo, ID 00 

De Varona, A. Sewer Gases. (Science Series No. 55.)... i6mo, o 50 
Devey, R. G. Mill and Factory Wiring. (Installation Manuals 

Series.) i2mo, *i 00 

Dibdin, W. J. Purification of Sewage and Water Svo, 6 50 

Dichman, C. Basic Open-Hearth Steel Process Svo, *3 50 

Dieterich, K. Analysis of Resins, Balsams, and Gum Resins 

Svo, *3 75 

Dilworth, E. C. Sfeel Railway Bridges 4to, *4 00 

Dinger, Lieut. H. C. Care and Operation of Naval Machinery 

i2mo. 3 00 

Dixon, D. B. Machinist's and Steam Engineer's Practical Cal- 
culator i6mo, mor., i 25 

Dodge, G. F. Diagrams for Designing Reinforced Concrete 

Structures folio, *4 00 

Dommett, W. E. Motor Car Mechanism i2mo, *2 25 

Dorr, B- F. The Surveyor's Guide and Pocket Table-book. 

i6mo, mor., 2 00 



D. VAN NOSTRAND COMPANY'S SHORT-TITLE CATALOG I3 

Draper, C. H. Elementary Text-book of Light, Heat and 

Sound i2mo, i 00 

Draper, C. H. Heat and the Principles of Thermo-dynamics, 

i2mo, 2 00 

Dron, R. W. Mining Formulas. lamo, i 00 

Dubbel, H. High Power Gas Engines 8vo, *5 00 

Dumesny, P., and Noyer, J. Wood Products, Distillates, and 

Extracts 8vo, *6 25 

Duncan, W. G., and Penman, D. The Electrical Equipment of 

Collieries 8vo, *6 75 

Dunkley, W. G. Design of Machine Elements 2 vols. 

i2mo, each, *2 50 
Dunstan, A. E., and Thole, F. B. T. Textbook of Practical 

Chemistry i2mo, *i 40 

Durham, H. W. Saws Bvo, 2 50 

Duthie, A. L. Decorative Glass Processes. (Westminster 

Series) 8vo, *2 00 

Dwight, H. B. Transmission Line Formulas 8vo *2 00 

Dyson, S. S. Practical Testing of Raw Materials , 8vo, *5 00 

and Clarkson, S. S. Chemical Works 8vo, 9 00 

Eccles, W. H. Wireless Telegraphy and Telephony. .i2mo, *8 80 
Eck, J. Light, Radiation and Illumination. Trans, by Paul 

Hogner 8vo, *2 50 

Eddy, H. T. Maximum Stresses under Concentrated Loads, 

8vo, I 50 
Eddy, L. C. Laboratory Manual of Alternating Currents, 

i2mo, o 50 

Edelman, P. Inventions and Patents i2mo *i 50 

Edgcumbe, K. Industrial Electrical Measuring Instruments . 

8vo, (In Press.) 
Edler, R. Switches and Switchgear. Trans, by Ph. Laubach. 

8vo, *4 00 

Eissler, M. The Metallurgy of Gold 8vo, 9 00 

The Metallurgy of Silver 8vo, 4 00 

The Metallurgy of Argentiferous Lead 8vo, 6 25 

— — A Handbook of Modern Explosives 8vo, 5 00 

Ekin, T. C. Water Pipe and Sewage Discharge Diagrams 

folio, *3 00 



14 D. VAN NOSTRAND COMpANY's SHORT-TITLE CATALOG 

Electric Light Carbons, Manufacture of 8vo, i oo 

Eliot, C. W., and Storer, F. H. Compendious Manual of Qualita- 
tive Chemical Analysis i2mo, *i 25 

Ellis, C. Hydrogenation of Oils 8vo, (In Press.) 

Ellis, G. Modern Technical Drawing 8vo, *2 00 

Ennis, Wm. D. Linseed Oil and Other Seed Oils 8vo, *4 00 

Applied Thermodynamics 8vo, *4 50 

Flying Machines To-day i2mo, *i 50 

Vapors for Heat Engines i2mo, *i 00 

Ermen, W. F. A. Materials Used in Sizing Svo, *2 00 

Erwin, M. The Universe and the Atom i2mo, *2 00 

Evans, C. A. Macadamized Roads {In Press.) 

Ewing, A. J. Magnetic Induction in Iron Svo, *4 00 

Fairie, J. Notes on Lead Ores i2mo, *o 50 

— — Notes on Pottery Clays i2mo, *2 25 

Fairley, W., and Andre, Geo. J. Ver-tilation of Coal Mines. 

(Science Series No. 58.) i6mo, o 50 

Fairweather, W. C. Foreign and Colonial Patent Laws . . .8vo, *3 00 

Falk, M. S. Cement Mortars and Concretes Svo, *2 50 

Fanning, J. T. Hydraulic and Water-supply Engineering . Svo, *5 00 

Fay, I. W. The Coal-tar Colors Svo, *4 00 

Fernbach, R. L. Glue and Gelatine Svo, *3 00 

Findlay, A. The Treasures of Coal Tar. i2mo, 2 00 

Firth, J. B. Practical Physical Chemistry. i2mo, *i 25 

Fischer, E. The Preparation of Organic Compounds. Trans. 

by R. V. Stanford i2mo, *i 25 

Fish, J. C. L. Lettering of Working Drawings. . . .Oblong Svo, i 00 

■ Mathematics of the Paper Location of a Railroad, 

i2mo, paper, *o 25 
Fisher, H. K. C, and Darby, W. C. Submarine Cable Testing. 

Svo, *3 50 
Fleischmann, W. The Book of the Dairy. Trans, by C. M. 

Aikman Svo, 4 50 

Fleming, J. A, The Alternate-current Transformer. Two 

Volumes Svo, 

VoL I. The Induction of Electric Currents *5 50 

Vol. II. The Utilization of Induced Currents *5 50 



D. VAN NOSTRAND COMPANY S SHORT-TITLE CATALOG 1 5 

Fleming, J. A. Propagation of Electric Currents 8vo, *3 oo 

A Handbook for the Electrical Laboratory and Testing 

Room. Two Volumes 8vo, each, *5 oo 

Fleury, P. Preparation and Uses of White Zinc Paints. .8vo, *3 50 

Flynn, P. J. Flow of Water. (Science Series No. 84.) .lamo, o 50 

Hydraulic Tables. (Science Series No. 66.) i6mo, o 50 

Forgie, J. Shield Tunneling Svo. {In Press.) 

Foster, H. A. Electrical Engineers' Pocket-book. {Seventh 

Edition.) 1 2mo, leather, 5 00 

Engineering Valuation of Public Utilities and Factories, 

Svo, *3 00 

Handbook of Electrical Cost Data Svo. {In Press) 

Fowls, F. F. Overhead Transmission Line Crossings .... i2mo, *i 50 

The Solution of Alternating Current Problems Svo {In Press.) 

Fox, W. G. Transition Curves. (Science Series No. no.). i6mo, o 50 
Fox, W., and Thomas, C. W. Practical Course in Mechanical 

Drawing i2mo, i 25 

Foye, J. C. Chemical Problems. (Science Series No. 69.). i6mo, o 50 

Handbook of Mineralogy. (Science Series No. 86.) . 

i6mo, o 50 

Francis, J. B. Lowell Hydraulic Experiments 4to, 15 00 

Franzen, H. Exercises in Gas Analysis i2mo, *i 00 

Freudemacher, P. W. Electrical Mining Installations. (In- 
stallation Manuals Series.) i2mo, *i 00 

Friend, J. N. The Chemistry of Linseed Oil i2mo, i 00 

Frith, J. Alternating Current Design 8vo, *2 50 

Fritsch, J. Manufacture of Chemical Manures. Trans, by 

D. Grant Svo, *6 50 

Frye, A. I. Civil Engineers' Pocket-book i2mo, leather, *5 00 

Fuller, G. W. Investigations into the Purification of the Ohio 

River 4to, *io 00 

Furnell, J. Paints, Colors, Oils, and Varnishes 8vo, 

Gairdner, J. W. I. Earthwork Svo (/n Press.) 

Gant, L. W. Elements of Electric Traction Svo, *2 50 

Garcia, A. T. R. V. Spanish-English Railway Terms. . . .Svo, *4 50 
Gardner, 5. A. Paint Researches and Their Practical 

Application 8vo, *5 00 



l6 D. VAN NOSTRAND COMPANY'S SHORT-TITLE CATALOG 

Garforth, W E. Rules for Recovering Coal Mines after Explo- 
sions and Fires i2mo, leather, i 50 

Garrard, C. C. Electric Switch and Controlling Gear 8vo, *6 00 

Gaudard, J. Foundations. (Science Series No. 34.) i6mo, o 50 

Gear, H. B., and Williams, P. F. Electric Central Station Dis- 
tributing Systems 8vo, 3 00 

Geerligs, H. C. P. Cane Sugar and Its Manufacture 8vo, *6 00 

Chemical Control in Cane Sugar Factories 4to, 5 00 

Geikie, J. Structural and Field Geology 8vo, *4 00 

Mountains, Their Origin, Growth and Decay Svo, *4 00 

The Antiquity of Man in Europe Svo, *3 00 

Georgi, F., and Schubert, A. Sheet Metal Working. Trans. 

by C. Salter Svo, 4 25 

Gerhard, W. P. Sanitation, Water-supply and Sewage Dis- 
posal of Country Houses lamo, *2 00 

Gas Lighting. (Science Series No. in.) i6mo, 50 

Gerhard, W. P. Household Wastes. (Science Series No. 97.) 

i6mo, o 50 

House Drainage. (Science Series No. 63.) i6mo o 50 

Sanitary Drainage of Buildings. (Science Series No. 93.) 

i6mo, o 50 

Gerhardi, C. W. H. Electricity Meters Svo, *6 00 

Geschwind, L. Manufacture of Alum and Sulphates. Trans. 

by C. Salter Svo, *5 00 

Gibbings, A. H. Oil Fuel Equipment for Locomotives . . . Svo, *2 50 

Gibbs, W. E. Lighting by Acetylene i2mo, *i 50 

Gibson, A. H. Hydraulics and Its Application .8vo, *$ 00 

Water Hammer in Hydraulic Pipe Lines i2mo, *2 00 

Gibson, A. H., and Ritchie, E. G. Circular Arc Bow Girder. 4to, *3 50 

Gilbreth, F. B. Motion Study lamo, *2 00 

Bricklaying System Svo, *3 00 

Field System i2mo, leather, *3 00 

Primer of Scientific Management i2mo, *i 00 

Gillette, H. P. Handbook of Cost Data. . . . . . .i2mo, leather, *5 00 

Rock Excavation Methods and Cost i2mo, leather, *5 00 

Handbook of Earth Excavation (In Press.) 

Handbook of Tunnels and Shafts, Cost and Methods 

of Construction (In Press.) 

Handbook of Road Construction, Methods and Cost. .(In Press.) 



D. VAN NOSTRAND COMPANY S SHORT-TITLE CATALOG 1/ 

Gillette, H. P., and Dana, R. T. Cost Keeping and Manage- 
ment Engineering 8vo, 

and Hill, C. S. Concrete Construction, Methods and 

Cost 8vo, 

Gillmore, Gen. Q. A. Roads, Streets, and Pavements.. .lamo, 
Godfrey, E. Tables for Structural Engineers. .i6mo, leather, 

Golding, H. A. The Theta-Phi Diagram i2mo, 

Goldschmidt, R. Alternating Current Commutator Motor . 8vo, 

Goodchild, W. Precious Stones. (Westminster Series.) .8vo, 

Goodell, J. M. The Location, Construction and Maintenance 

of Roads 8vo, 

Goodeve, T. M. Textbook on the Steam-engine i2mo, 

Gore, G. Electrolytic Separation of Metals 8vo, 

Gould, E. S. Arithmetic of the Steam-engine i2mo, 

Calculus. (Science Series No. 112.) i6mo, 

High Masonry Dams. (Science Series No. 22.) . . . i6mo, 

Practical Hydrostatics and Hydrostatic Formulas. (Science 

Series No. 117.) i6mo, 

Gratacap, L. P. A Popular Guide to Minerals Svo, 

Gray, J. Electrical Influence Machines i2mo, 

Gray, J. Marine Boiler Design - - . . i2mo, 

Greenhill, G. Dynamics of Mechanical Flight Svo. 

Gregorius, R. Mineral Waxes. Trans, by C. Salter. .. lamo, 
Grierson, R. Modern Methods of Ventilation Svo, 

Griffiths, A. B. A Treatise on Manures i2mo, 

Griffiths, A. B. Dental Metallurgy Svo, 

Gross, E. Hops Svo, *6 25 

Grossman, J. Ammonia and its Con'r)ounds i2mo, *i 25 

Groth, L. A. Welding and Cutting Metals by Gases or Electric- 
ity. (Westminster Series.) Svo, *2 00 

Grover, F. Modern Gas and Oil Engines Svo, *3 00 

Gruner, A. Power-loom Weaving Svo, *3 00 

Grunsky, C. E. Topographic Stadia Surveying i2mo, 2 00 

GUldner, Hugo. Internal-Combustion Engines. Trans, by 

H. Diedrichs. , 4to, 15 00 



*3 


50 


*5 


GO 


I 


25 


*2 


50 


*2 


00 


*3 


00 


*2 


00 


I 


50 


2 


00 


*3 


50 


I 


00 





50 





50 





50 


*2 


GO 


2 


00 


*I 


25 


*2 


50 


*3 


50 


3 


GO 


3 


00 


*4 


25 



l8 D. VAN NOSTRAND COMPANY'S SHORT-TITLE CATALOG 

Gunther, C. 0. Integration 8vo, *i 25 

Gurden, R. L. Traverse Tables folio, half mor., *7 50 

Guy, A. E. Experiments on the Flexure of Beams 8vo, *i 25 

Haenig, A. Emery and the Emery Industry 8vo, *3 00 

Hainbach, R. Pottery Decoration. Trans, by C. Salter. i2mo, *4 25 

Hale, W. J. Calculations of General Chemistry i2mo, *i 25 

Hall, C. H. Chemistry of Paints and Paint Vehicles. .... i2mo, *2 00 
Hall, G. L. Elementary Theory of Alternate Current Work- 
ing 8vo, 

Hall, R. H. Governors and Governing Mechanism 1200, 

Hall, W. S. Elements of the Differential and Integral Calculus 

8vo, 

■ Descriptive Geometry . 8vo volume and 4to atlas, 

Haller, G. F., and Cunningham, E. T. The Tesla Coil i2mo, 

Halsey, F. A Slide Valve Gears i2mo, 

The Use of the Slide Rule. (Science Series No. 114.) 

i6mo, 
Worm and Spiral Gearing. (Science Series No. 116.) 

i6mo, 

Hancock, H. Textbook of Mechanics and Hydrostatics. ... Svo, 
Hancock, W. C. Refractory Materials. (Metallurgy Senes.(In Press.) 

Hardy, E. Elementary Principles of Graphic Statics i2mo, *i 50 

Haring, H. Engineering Law. 

Vol. I. Law of Contract Svo, *4 00 

Harper, J. H. Hydraulic Tables on the Flow of Water. i6mo, *2 00 

Harris, S. M. Practical Topographical Surveying (In Press.) 

Harrison, W. B. The Mechanics' Tool-book i2rno, ^^ i 50 

Hart, J. W. External Plumbing Work Svo, ^3 25 

Hints to Plumbers on Joint Wiping Svo, *4 25 

— — Principles of Hot Water Supply Svo, *4 25 

Sanitary Plumbing and Drainage Svo, *4 25 

Haskins, C. H. The Galvanometer and Its Uses i6mo, i 50 

Hatt, J. A. H. The Colorist square i2mo, *i 50 

Hausbrand, E. Drying by Means of Air and Steam. Trans. 

by A. C. Wright i2mo, *3 00 

Evaporating, Condensing and Cooling Apparatus. Trans. 

by A. C, Wright Svo, *7 25 



*2 


50 


*2 


25 


*3 


50 


*i 


25 


I 


50 





50 





50 


I 


50 



b. VAN NOSTRAND COMPANY S SHORT-TITLE CATALOG I9 

Hausmann, E. Telegraph Engineering .8vo, *3 00 

Hausner, A. Manufacture of Preserved Foods and Sweetmeats. 

Trans, by A. Morris and H. Robson 8vo, *4 25 

Hawkesworth, T. Graphical Handbook for Reinforced Concrete 

Design 4to, *2 50 

Hay, A. Continuous Current Engineering 8vo, *2 50 

Hayes, H. V. Public Utilities, Their Cost New and Deprecia- 
tion Svo, *2 00 

Public Utilities, Their Fair Present Value and Return, 

Svo, *2 00 

Heath, F. H. Chemistry of Photography.. Svo {In Press.) 

Heather, H. J. S. Electrical Engineering Svo, *3 50 

Heaviside, 0. Electromagnetic Theory. 

Vols. I and II Svo, each, *5 00 

Vol. Ill Svo, *7 50 

Heck, R. C. H. Steam Engine and Turbine Svo, *3 50 

Steam-Engine and Other Steam Motors. Two Volumes. 

Vol. I. Thermodynamics and the Mechanics Svo, *3 50 

Vol. II. Form, Construction and Working Svo, *5 00 

Notes on Elementary Kinematics Svo, boards, *i 00 

Graphics of Machine Forces Svo, boards, * i 00 

Heermann, P. Dyers' Materials. Trans, by A. C. Wright. 

i2mo, *3 00 
Heidenreich, E. L. Engineers' Pocketbook of Reinforced 

Concrete i6mo, leather, *3 00 

Hellot, Macquer and D'Apligny. Art of Dyeing Wool, Silk and 

Cotton Svo, *2 00 

Henrici, 0. Skeleton Structures Svo, i 50 

Hering, C, and Getmann, F. H. Standard Tables of Electro- 
chemical Equivalents *2 00 

Hering, D. W. Essentials of Physics for College Students. 

Svo, *i 75 
Hering-Shaw, A. Domestic Sanitation and Plumbing. Two 

Vols Svo, *5 00 

Elementary Science Svo, *2 00 

Herington, C. F. Powdered Coal as a Fuel Svo, 3 00 

Hermann, G. The Graphical Statics of Mechanism. Trans. 

by A. P. Smith lamo, 2 00 

Herzfeld, J. Testing of Yarns and Textile Fabrics Svo, *6 25 



20 D. VAN NOSTRAND COMPANY S SHORT-TITLE CATALOG 

Hildebrandt, A. Airships, Past and Present 8vo, 

Hildenbrand, B. W. Cable-Making. (Science Series No. 32.) 

i6mo, o 50 

Hilditch, T. P. Concise History of Chemistry lamo, *i 25 

Hill, C. S. Concrete Inspection , i6mo, *i 00 

Hill, C. W. Laboratory Manual and Notes in Beginning 

Chemistry {In Press.) 

Hill, J. W. The Purification of Public Water Supplies, New 

Edition {In Press ) 

Interpretation of Water Analysis {In Press ) 

Hill, M. J. M. The Theory of Proportion ^. 8vo, *2 50 

Hiroi, I. Plate Girder Construction. (Science Series No. 95.) 

i6mo, o 50 

Statically-Indeterminate Stresses i2mo, *2 00 

Hirshfeld, C. F. Engineering Thermodynamics. (Science 

Series No. 45) i6mo, o 50 

Hoar, A. The Submarine Torpedo Boat lamo, *2 00 

Hobart, H. M. Heavy Electrical Engineering 8vo, *4 50 

Design of Static Transformers i2mo, *2 00 

Electricity 8vo, *2 00 

Electric Trains.. - Svo, *2 50 

Electric Propulsion of Ships .8vo, *2 50 

Hobart, J. F. Hard Soldering, Soft Soldering, and Brazing 

i2mo, *i 00 
Hobbs, W. R. P. The Arithmetic of Electrical Measurements 

i2mo, o 75 

Hoff, J. N. Paint and Varnish Facts and Formulas . . . f2mo. *i 50 

Hole, W. The Distribution of G-as Svo, *8 50 

HoUey, A. L. Railway Practice. folio, 6 00 

Hopkins, N. M. Model Engines and Small Boats i2mo, i 25 

Hopkinson, J., Shoolbred, J. N., and Day. R. E Dynamic 

Electricity. (Science Series No, 71.) i6m0; o 50 

Horner, J. Practical Ironfounding Svo, *2 00 

Gear Cutting, in Theory and Practice Svo, *3 00 

Horniman,. R. How to Make the Railways Pay for the War, 

i2mo, 3 CO 

Houghton, C. E. The Elements of Mechanics of Materials. i2mo, *2 00 

Houstoun, R. A. Studies in Light Production i2mo, 2 00 



D. VAN NOSTRAND COMPANY'S SHORT-TITLE CATALOG 21 

Hovenden, F. Practical Mathematics for Young Engineers, 



*i 



i2mo, ""I 50 

Howe, G. Mathematics for the Practical Man. - i2mo, *i 23 

Howorth, J. Repairing and Riveting Glass, China and Earthen- 
ware .8vo, paper, *i 00 

Hoyt, W. F. Chemistry by Experimentation i2mo, *o 70 

Hubbard, E. The Utilization of Wood-waste 8vc, *3 00 

Hubner, J. Bleaching and Dyeing of Vegetable and Fibrous 

Materials. (Outlines of Industrial Chemistry.) .8vo, *5 00 

Hudson, O. F. Iron and Steel. (Outlines of Industrial 

Chemistry.) Svo, *2 00 

Humphrey, J. C. W. Metallography of Strain. (Metallurgy 

Series) (In Press.) 

Humphreys, A. C. The Business Features of Engineering 

Practice Svo, *i 25 

Hunter, A. Bridge Work Svo (In Press.) 

Hurst, G. H. Handbook of the Theory of Color Svo, *4 25 

Dictionary of Chemicals and Raw Products Svo, *6 25 

t Lubricating Oils, Fats and Greases Svo, *7 25 

Soaps Svo, *7 25 

Hurst, G. H., and Simmons, W. H. Textile Soaps and Oils, 

Svo, 4 25 

Hurst, H. E., and Lattey, R. T. Text-book of Physics Svo, *3 00 

'■ Also published in Three Parts: 

Vol. I. Dynamics and Heat *i 25 

Vol. II. Sound and Light *i 25 

Vol. III. Magnetism and Electricity *i 50 

Hutchinson, R. W., Jr. Long Distance Electric Power Trans- 
mission i2mo, *3 00 

Hutchinson, R. W., Jr., and Thomas, W. A. Electricity in 

Mining (In Press.) 

Hutchinson, W. B. Patents and How to Make Money Out of 

Them i2mo, i 00 

Hutton, W. S. The Works' Manager's Handbook. Svo, 6 00 

Hyde, E. W. Skew Arches. (Science Series No. 15.).. . . i6mo, o 50 

Hyde, F. S. Solvents, Oils, Gums and Waxes Svo, *2 00 



22 D. VAN NOStflANt) COMPANY S SHORt-TlTLE CATALOG 

Induction Coils. (Science Series No. 53.) i6mo, o 50 

Ingham, A. E. Gearing. A practical treatise 8vo, *2 50 

Ingle, H. Manual of Agricultural Chemistry 8vo, *4 25 

Inness, C. H. Problems in Machine Design lamo, *3 00 

Air Compressors and Blowing Engines i2mo, 

— — Centrifugal Pumps lamo, *3 00 

The Fan i2mo, *4 00 

Jacob, A., and Gould, E. S. On the Designing and Construction 

of Storage Reservoirs. (Science Series No. 6.). .i6mo, 50 
Jannettaz, E. Guide to the Determination of Rocks. Trans. 

by G. W. Plympton i2mo, i 50 

Jehl, F. Manufacture of Carbons 8vo, *4 00 

Jennings, A. S. Commercial Paints and Painting. (West- 
minster Series.) 8vo, *4 00 

Jennison, F. H. The Manufacture of Lake Pigments 8vo, *3 00 

Jepson, G. Camsand the Principles of their Construction. . .8vo, *i 50 

Mechanical Drawing 8vo {In Preparation.) 

Jervis-Smith, F. J. Dynamometers Svo, *3 50 

Jockin, W. Arithmetic of the Gold and Silversmith. . . . i2mo, *i 00 
Johnson, J, H. Arc Lamps and Accessory Apparatus. (In- 
stallation Manuals Series.) i2mo, *o 75 

Johnson, T. M. Ship Wiring and Fitting. (Installation 

Manuals Series.) i2mo, *o 75 

Johnson, W. McA. The Metallurgy of Nickel {In Preparation.) 

Johnston, J. F. W., and Cameron, C. Elements of Agricultural 

Chemistry and Geology. i2mo, 2 60 

Joly, J. Radioactivity and Geology. i2mo, *3 00 

Jones, H. C. Electrical Nature of Matter and Radioactivity 

i2mo, *2 CO 

Nature of Solution Svo, *3 50 

New Era in Chemistry i2mo, *2 00 

Jones, J. H, Tinplate Industry Svo, *3 00 

Jones, M. W. Testing Raw Materials Used in Paint. .. .i2mo, *3 00 

Jordan, L. C. Practical Railway Spiral lamo. Leather, *i 50 



D. VAN NOSTRAND COMPANY S SHORT-TITLE CATALOG 23 

Joynson, F. H. Dwigning and Construction of Machine Gear- 
ing 8vo, 2 00 

Juptner, H. F. V. Siderology: The Science of Iron 8vo, *6 25 



KapP) G. Alternate Current Machinery. (Science Series No. 

96.) i6mo, 50 

Kapper, F. Overhead Transmission Lines 4to, *4 00 

Keim, A. W. Prevention of Dampness in Buildings 8vo, *3 00 

Keller, S. S. Mathematics for Engineering Students. 

i2mo, half leather, 

and Knox, W. E. Analytical Geometry and Calculus.. *2 00 

Kelsey, W. R. Continuous-current Dynamos and Motors. 

8vo, *2 50 
Kemble, W. T., and Underhill, C. R. The Periodic Law and the 

Hydrogen Spectrum 8vo, paper, 

Kemp, J. F. Handbook of Rocks Svo, 

Kendall, E. Twelve Figure Cipher Code 4to, 

Kennedy, A. B. W., and Thurston, R. H. Kinematics of 

Machinery. (Science Series No. 54.) i6mo, o 50 

Kennedy, A. B. W., Unwin, W. C, and Idell, F. E. Compressed 

Air. (Science Series No. 106.) i6mo, 

Kennedy, R. Electrical Installations. Five Volumes 4to, 

Single Volumes each, 

Fljang Machines ; Practice and Design lamo, 

Principles of Aeroplane Construction Svo, 

Kennelly, A. E. Electro-dynamic Machinery Svo, 

Kent, W. Strength of Materials. (Science Series No. 41.), i6mo, 

Kershaw, J. B. C. Fuel, Water and Gas Analysis Svo, 

Electrometallurgy. (Westminster Series.) Svo, 

. The Electric Furnace in Iron and Steel Production. . i2mo, 

Electro-Thermal Methods of Iron and Steel Production, 

Svo, 
Kindelan, J. Trackman's Helper. . .- lamo, 2 00 



*0 


50 


*I 


50 


*I2 


50 






50 


15 


00 


3 


50 


*2 


50 


*2 


00 


I 


50 





50 


*2 


50 


*2 


00 


*3 


00 



24 D. VAN NOSTRAND COMPANY S SHORT-TITLE CATALOG 



Kinzbrunner, C. Alternate Current Windings 8vo, *I 50 

Continuous Current Armatures 8vo, *i 50 

Testing of Alternating Current Machines 8vo, *2 oc 

Kirkaldy, A. W., and Evans, A. D. History and Economics 

of Transport 8vo, *3 00 

Kirkaldy, W. G. David Kirkaldy's System of Mechanical 

Testing 4to, 10 00 

Kirkbride, J. Engraving for Illustration 8vo, *i 75 

Kirkham, J. E. Structural Engineering 8vo, *5 00 

Kirkwood, J. P. Filtration of River Waters 4to, 7 50 

Kirschke, A. Gas and Oil Engines i2mo, *i 50 

Klein, J. F. Design of a High speed Steam-engine Svo, *5 00 

Physical Significance of Entropy Svo, *i 50 

Klir^genberg, G. Large Electric Power Stations 4to, *5 00 

Knight, R.-Adm. A. M. Modem Seamanship Svo, *6 50 

Pocket Edition fabrikoid, i2mo, 3 00 

Knott, C. G., and Mackay, J. S. Practical Mathematics. . .Svo, 2 00 

Knox, G. D. Spirit of the Soil lamo, *i 25 

Fi-'nox, J. Physico-chemical Calculations i2mo. *i 25 

Fixation of Atmospheric Nitrogen. (Chemical Mono- 
graphs.) i2mo, I 00 

Koester, F. Steam-Electric Power Plants 4to, *5 00 

Hydroelectric Developments and Engineering 4to, ''5 00 

KoUer, T. The Utilization of Waste Products Svo, *6 50 

Cosmetics Svo, *3 00 

Koppe, S. W. Glycerine i2mo, '''4 25 

Kozmin, P. A. Flour Milling. Trans, by M. Falkner. .Svo, 7 50 
Kremann, R. Application of Phydco Chemical Theory to 
Technical Processes and Manufacturing Methods. 

Trans, by H. E. Potts Svo, ^3 00 

Kretchmar, K. Yarn and Warp Sizing Svo, *6 25 

Laffargue, A. The Attack in Trench Warfare 32mo, o 50 

Lallier, E. V. Elementary Manual of the Steam Engine. 

I2mO, *2 GO 

Lambert, T. Lead and its Compounds Svo, "^425 

Bone Products and Manures Svo, "^4 25 



D. VAN NOSTRAND COMPANY S SHORT-TITLE CATALOG 25 

Lamborn, L. L. Cottonseed Products 8vo, *3 oo 

Modern Soaps, Candles, and Glycerin 8vo, *7 50 

Lamprecht, R. Recovery Work After Pit Fires. Trans, by 

C. Salter " 8vo, *6 25 

Lancaster, M. Electric Cooking, Heating and Cleaning. .8vo, *i 00 
Lanchester, F. W. Aerial Flight. Two Volumes. Svo. 

Vol. II. Aerodonetics *6 00 

Vol. I. Aerodynamics *6 00 

— The Flying Machine Svo, *3 00 

Industrial Engineering: Present and Post-War Outlook, 

i2mo, I 00 

Lange, K. R. By-Products of Coal-Gas Manufactuie. .lamo, 3 00 

Larner, E. T. Principles of Alternating Currents lamo, *i 25 

La Rue, B. F. Swing Bridges. (Science Series No. 107.) . i6mo, o 50 
Lassar-Cohn, Dr. Modern Scientific Chemistry. Trans, by M. 

M. Pattison Mtiir 1 2mo, *2 00 

Latimer, L. H., Field, Co J., and Howell, J. W. Incandescent 

Electric Lighting. (Science Series No. 57.) i6mo, o 50 

Latta, M. N. Handbook of American Gas-Engineering Practice. 

Svo, *4 50 

American Producer Gas Practice . 4to, *6 00 

Laws, B. C. Stability and Equilibrium of Floating Bodies. Svo, *3 50 
Lawson, W. R. British Railways, a Financial and Commer- 
cial Survey Svo, 2 00 

Leask, A. R. Breakdowns at Sea i2mo, 2 00 

Refrigerating Machinery . i2mo, 2 00 

Lecky, S. T. S. "Wrinkles" in Practical Navigation .... Svo, *io 00 
Danger Angle i6mo, 2 50 

Le Doux, M. Ice -Making Machines. (Science Series No. 46.) 

i6mo, o 50 

Leeds, C. C. Mechanical Drawing for Trade Schools. oblong 4to, *2 00 

Mechanical Drawing for High and Vocational Schools, 

4to, *i 25 
Lef^vre, L. Architectural Pottery. Trans, by H. K. Bird and 

W. M. Birns 4to, *8 50 

Lehner, S. Ink Manufacture. Trans, by A. Morris and H 

Robson Svo, ^'3 00 

Lemstrom, S. Electricity in Agriculture and Horticulture. . Svo, *i 50 

Letts, E. A. Fundamental Problems in Chemistry Svo, *2 00 



26 D. VAN NOSTRAND COMPANY'S SHORT-TITLE CATALOG 

Le Van, W. B. Steam-Engine Indicator. (Science Series No. 

78.) i6mo, o 50 

Lewes, V.B. Liquid and Gaseous Fuels. (Westminster Series.) 

8V0, *2 GO 

Carbonization of Coal 8vo, *4 00 

Lewis Automatic Machine Rifle ; Operation of . . i6mo, *o 60 

Lewis, L. P. Railway Signal Engineering 8vo, *3 50 

Licks, H. E. Recreations in Mathematics izmo, 1 25 

Lieber, B. F. Lieber's Five Letter Standard Telegraphic Code, 

8vo, *i5 00 

Spanish Edition Svo, *i5 00 

French Edition Svo, *i5 00 

Terminal Index Svo, *2 50 

Lieber's Appendix folio, *i5 00 

Handy Tables 4to, *2 50 

Bankers and Stockbrokers' Code and Merchants and 

Shippers' Blank Tables Svo, *i5 00 

Lieber, B. F. 100,000,000 Combination Code Svo, *io 00 

Engineering Code Svo, *i2 50 

Livermore, V. P., and Williams, J. Ho^ to Become a Com- 
petent Motorman i2mo, *i 00 

Livingstone, R. Design and Construction of Commutators . Svo, *2 25 

Mechanical Design and Construction of Generators. ..Svo, *3 50 

Lloyd, S. L. Fertilizer Materials 2 00 

Lobben, P. Machinists' and Draftsmen's Handbook Svo, 2 50 

Lockwood, T. D. Electricity, Magnetism, and Electro-teleg- 
raphy Svo, 2 50 

Electrical Measurement and the Galvanometer i2mo, o 75 

Lodge, 0. J. Elementary Mechanics. i2mo, i 5c 

Signalling Across Space without Wires Svo, *2 00 

Loewenstein, L. C, and Crissey, C. P. Centrifugal Pumps. . *4 50 

Lomax, J. W. Cotton Spinning i2mo, i 50 

Lord, R. T. Decorative and Fancy Fabrics Svo, *4 25 

Loring, A. E. A Handbook of the Electromagnetic Telegraph, 

i6mo, o 50 

Handbook. (Science Series No. 39) i6mo, o 50 

Lovell, D. H. Practical Switchwork. Revised by Strong and 

Whitney (In Press.) 



D. VAN NOSTRAND COMPANY'S SHORT-TITLE CATALOG 2'J 

Low, D. A. Applied Mechanics (Elementary) i6mo, o 80 

Lubschez, B. J. Perspective i2mo, *i 50 

Lucke, C. E. Gas Engine Design 8vo, *3 00 

Power Plants: their Design, Efficiency, and Power Costs. 

2 vols {In Preparation.) 

Luckiesh, M. Color and Its Application 8vo, *3 00 

Light and Shade and Their Applications Svo, *2 50 

The two when purchased together *5 00 

Lunge, G. Coal-tar Ammonia. Three Parts Svo, *25 00 

Manufacture of Sulphuric Acid and Alkali. Four Volumes. 

Svo, 

Vol. I. Sulphuric Acid. In three parts *i8 00 

Vol. I. Supplement 5 00 

Vol. II. Salt Cake, Hydrochloric Acid and Leblanc Soda. 

In two parts {In Press.) 

Vol. III. Ammonia Soda (In Press.) 

Vol. rV. Electrolytic Methods (In Press.) 

Technical Chemists' Handbook i2mo, leather, *4 00 

Technical Methods of Chemical Analysis. Trans, by 

C. A. Keane. In collaboration with the corps of 
specialists. 

VoL I. In two parts Svo, *i5 00 

Vol. II. In two parts Svo, *i8 00 

Vol. III. In two parts Svo, *iS 00 

The set (3 vols.) complete *5o 00 

Technical Gas Analysis Svo, *4 50 

Luquer, L. M, Minerals in Rock Sections Svo, *i 50 

MacBride, J. D. A Handbook of Practical Shipbuilding, 

i2mo, fabrikoid 2 00 

Macewen, H. A. Food Inspection Svo, *2 50 

Mackenzie, N. F. Notes on Irrigation Works Svo, *2 50 

Mackie, J. How to Make a Woolen Mill Pay Svo, *2 25 

Maguire, Wm. R. Domestic Sanitary Drainage and Plumbing 

Svo, 4 00 

Malcolm, C. W. Textbook on Graphic Statics Svo, *3 00 

Malcolm, H. W. Suomarine Telegraph Cable (In Press.) 

Mallet, A. Compound Engines. Trans, by R. R. Buel. 
(Science Series No. 10.) i6mo, 



28 D. VAN NOSTRAND COMPANY'S SHORT-TITLE CATALOG 

Mansfield, A. N. Electro-magnets. (Science Series No. 64) 

i6mo, o 50 
Marks, E. C. R. Construction of Cranes and Lifting Machinery 

i2mo, *2 00 

Construction and Working of Pumps i2mo, 

Manufacture of Iron and Steel Tubes i2mo, *2 00 

Mechanical Engineering Materials i2mo, *i 50 

Marks, G. C. Hydraulic Power Engineering 8vo, 4 50 

Inventions, Patents and Designs i2mo, *i 00 

Marlow, T. G. Drying Machinery and Practice 8vo, *5 00 

Marsh, C. F. Concise Treatise on Reinforced Concrete . . 8vo, *2 50 

Marsh, C. F. Reinforced Concrete Compression Member 

Diagram Mounted on Cloth Boards *i 50 

Marsh, C. F., and Dunn, W. Manual of Reinforced Concrete 

and Concrete Block Construction. .. .i6mo, fabrikoid, 

(In Press.) 
Marshall, W.J., and Sankey, H. R. Gas Engines. (Westminster 

Series.) 8vo, *2 00 

Martin, G. Triumphs and Wonders of Modem Chemistry. 

Svo, *2 00 

Modern Chemistry and Its Wonders Svo, *2 00 

Martin, N. Properties and Design of Reinforced Concrete, 

i2mo, *2 50 

Martin, W. D. Hints to Engineers lamo, i 50 

Massie, W. W., and Underhill, C. R. Wireless Telegraphy and 

Telephony i2mo, *i 00 

Mathot, R. E. Internal Combustion Engines Svo, *4 00 

Maurice, W. Electric Blasting Apparatus and Explosives ..Svo, *3 50 

Shot Firer's Guide Svo, *i 50 

Maxwell, F. Sulphitation in White Sugar Manufacture. i2mo, 3 75 
Maxwell, J. C. Matter and Motion. (Science Series No. 36.) 

i6mo, o 50 
Maxwell, W. H., and Brown, J. T. Encyclopedia of Municipal 

and Sanitary Engineering 4to, *io 00 

Mayer, A. M. Lecture Notes on Physics Svo, 2 00 

Mayer, C, and Slippy, J. C. Telephone Line Construction . Svo, *3 00 

McCullough, E. Practical Surveying , .-. i2mo, *2 00 

Engineering Work in Cities and Towns., Svo, *3 00 

Reinforced Concrete i2mo, *i 50 

McCullough, R, S. Mechanical Theory of Heat.o Svo, 3 50 



D. VAN NOSTRAND COMPANY'S SHORT-TITLE CATALOG 29 

McGibbon, W. C. Indicator Diagrams for Marine Engineers, 

8vo, *3 50 

Marine Engineers' Drawing Book oblong 4to, *2 50 

Marine Engineers' Pocketbook lamo, leather, *4 00 

Mcintosh, J. G. Technology of Sugar 8vo, *7 25 

Industrial Alcohol 8vo, *4 25 

Manufacture of Varnishes and Kindred Industries. 

Three Volumes. Svo. 

Vol. I. Oil Crushing, Refining and Boiling 

Vol. II. Varnish Materials and Oil Varnish Making *6 25 

Vol. III. Spirit Varnishes and Materials *7 25 

McKay, C. W. Fundamental Principles of the Telephone 

Business Svo, (In Press.) 

McKillop, M., and McKillop, D. A. Efficiency Methods. 

i2mo, I 50 
McKnight, J. D., and Brown, A. W. Marine Multitubular 

Boilers *2 50 

McMaster, J. B. Bridge and Tunnel Centres. (Science Series 

No. 20.) i6mo, 50 

McMechen, F. L. Tests for Ores, Minerals and Metals. . . i2mo, *i 00 

McPherson, J. A. Water- works Distribution Svo, 2 50 

Meade, A. Modern Gas Works Practice Svo, *8 50 

Meade, R. K. Design and Equipment of Small Chemical 

Laboratories Svo, 

Melick, C. W. Dairy Laboratory Guide i2mo, *i 25 

Mensch, L. J. Reinforced Concrete Pocket Book.iGmo, leather *4 00 
"Mentor." Self -Instruction for Students in Gas Supply, 

i2mo, 2 50 

Advanced Self -Instruction for Students in Gas Supply, 

i2mo, 2 50 
Merck, E. Chemical Reagents: Their Purity and Tests. 

Trans, by H. E. Schenck Svo, i 00 

Merivale, J. H. Notes and Formulae for Mining Students, 

i2mo, I 50 
Merritt, Wm. H. Field Testing for Gold and Silver . i6mo, leather, 2 00 
Mertens, Colonel. Tactics and Technique in River Crossings. 

Translated by Major Walter Krueger Svo, 2 50 

Mierzinski, S. Waterproofing of Fabrics. Trans, by A. Morris 

and H. Robson Svo, *3 00 

Miessner, B. F. Radiodynamics i2mo, *2 00 

Miller, G. A. Determinants. (Science Series No. 105.). . i6mo, 

Miller, W. J. Historical Geology i2mo, *2 00 

Mills, C. N. Elementary Mechanics for Engineers i2mo, *i 00 

Milroy, M. E. W. Home Lace -making... i2mo, *i 00 



JO D. VAN NOSTRAND COMPANY S SHORT-TITLE CATALOG 

Mitchell, C. A. Mineral and Aerated Waters 8vo, *3 oo 

——and Prideaux, R. M. Fibres Used in Textile and 

AUied Industries 8vo, ^^4 25 

Mitchell, C. F. and G. A. Building Construction and Draw- 
ing i2mo 

Elementary Course, *i 50 
Advanced Course, *2 50 
Monckton, C. C. F. Radiotelegraphy. (Westminster Series.) 

8vo, *2 00 
Monteverde, R. D. Vest Pocket Glossary of English-Spanish, 

Spanish-English Technical Terms 64mo, leather, *i 00 

Montgomery, J. H. Electric Wiring Specifications. .. .i6mo, *i 00 
Moore, E. C. S. New Tables for the Complete Solution of 

Ganguillet and Kutter's Formula 8vo, *5 00 

Moore, Harold. Liquid Fuel for Internal Combustion Engines, 

8vo, 5 00 
Morecroft, J. H., and Hehre, F. W. Short Course in Electrical 

Testing 8vo, 

Morg.n, A. P. Wireless Telegraph Apparatus for Amateurs, 

i2mo, 
Morgan, C. E. Practical Seamanship for the Merchant 

Marine i2mo, fabrikoid {In Press.) 

Moses, A. J. The Characters of Crystals 8vo, 

and Parsons, C. L. Elements of Mineralogy Svo, 

Moss, S. A. Elements of Gas Engine Design. (Science 

Series No. 121) i6mo, 

The Lay-out of Corliss Valve Gears. (Science Series 

No. 119.) i6mo, 

Mulford, A. C. Boundaries and Landmarks i2mo, 

MuUin, J. P. Modern Moulding and Pattern-making. . . . i2mo,- 
Munby, A. E. Chemistry and Physics of Building Materials. 

(Westminster Series.) Svo, 

Murphy, J. G. Practical Mining i6mo, 

Murray, J. A. Soils and Manures. (Westminster Series.). Svo, 

Nasmith, J. The Student's Cotton Spinning Svo, 

■ Recent Cotton Mill Construction i2mo, 

Neave, G. B., and Heilbron, I. M. Identification of Organic 

Compounds i2mo, *i 25 

I^eilson, R. M. Aeroplane Patents Svo, *2 00 



*I 


50 


*I 


50 


res 


s.) 


*2 


00 


*3 50 





50 




*i 


50 
00 


2 


50 


*2 


00 


I 


00 


*2 


00 


3 


00 


2 


50 



D. VAN NOSTRAND COMPANY'S SHORT-TITLE CATALOG 3 1 

iNerz, F. Searchlights. Trans, by C. Rodgers 8vo, *3 oo 

Neuberger, H., and Noalhat, H. Technology of Petroleum. 

Trans, by J. G. Mcintosh 8vo, *io oo 

Newall, J. W. Drawing, Sizing and Cutting Bevel-gears. .8vo, i 50 
Newbigin, M. I., and Flett, J. S. James Geikie, the Man 

and the Geologist 8vo, 3 50 

Newbiging, T. Handbook for Gas Engineers and Managers, 

8vo, *6 50 
Newell, F. H., and Drayer, C. E. Engineering as a Career. 

121UO, cloth, *i 00 

paper, o 75 

Nicol, G. Ship Construction and Calculations 8vo, *5 00 

Nipher, F. E. Theory of Magnetic Measurements i2mo, r 00 

Nisbet, H. Grammar of Textile Design 8vo, 

Nolan, H. The Telescope. (Science Series No. 51.) i6mo, o 50 

Norie, J. W. Epitome of Navigation (2 Vols.) octavo, 15 00 

— > — A Complete Set of Nautical Tables with Explanations 

of Their Use octavo, 6 50 

North, H. B. Laboratory Experiments in General Chemistry 

i2mo, *i 00 

Nugent, E. Treatise on Optics i2mo, i 50 

O'Connor, H. The Gas Engineer's Pocketbook. .. i2mo, leathei, 3 50 
Ohm, G. S., and Lockwood, T. D. Galvanic Circuit. Trans, by 

William Francis. (Science Series No. 102.). . . .i6mo, o 50 

Olsen, J. C. Textbook of Quantitative Chemical Analysis. .Svo, *3 50 
Olsson, A. Motor Control, in Turret Turning and Gun Elevating. 

(U. S. Navy Electrical Series, No. i.) . ...i2mo, paper, *o 50 

Ormsby, M. T. M. Surveying i2mo, 2 50 

Oudin, M. A. Standard Polyphase Apparatus and Systems . .8vo, *3 00 

Owen, D. Recent Physical Research Svo, 

Fakes, W. C. C, and Nankivell, A. T. The Science of Hygiene. 

Svo, *i 75 
Palaz, A. Industrial Photometry. Trans, by G. W. Patterson, 

Jr Svo, *4 00 

Pamely, C. Colliery Manager's Handbook 8vo, *io 00 

Parker, P. A. M. The Control of Water Svo, *5 00 

Parr, G. D. A. Electrical Engineering Measuring Instruments. 

Svo, *3 50 
Parry, E. J. Chemistry of Essential Oils and Artificial Per- 
fumes 10 00 



32 D. VAN NOSTRAND COMPANY'S SHORT-TITLE CATALOG 

Parry, E J. Foods and Drugs. Two Volumes 8vo, 

Vol. I. Chemical and Microscopical Analysis of Food 

and Drugs *io oo 

Vol. II. Sale of Food and Drugs Acts *4 25 

and Coste, J. H. Chemistry of Pigments... 8vo, *6 50 

Parry, L. Notes on Alloys 8vo, *3 50 

Metalliferous Wastes .Svo, *2 50 

Analysis of Ashes and Alloys Svo, "^'2 50 

Parry, L. A. Risk and Dangers of Various Occupations. .Svo, *4 25 

Parshall, H. F., and Hobart, H. M. Armature Windings .... 4to, *7 50 

Electric Railway Engineering. 4to, *io 00 

Parsons, J. L. Land Drainage Svo, *i 50 

Parsons, S. J. Malleable Cast Iron Svo, *2 50 

Partington, J. R. Higher Mathematics for Chemical Students 

i2mo, *2 00 

Textbook of Thermodynamics Svo, *4 00 

Passmore, A. C. Technical Terms Used in Architecture . Svo, *4 25 

Patchell, W. H. Electric Power in Mines Svo, *4 00 

Faterson, G. W. L. Wiring Calculations. i2mo, *3 00 

Electric Mine Signalling Installations i2mo, *i 50 

Patterson, D. The Color Printing of Carpet Yarns Svo, *4 25 

' Color Matching on Textiles Svo, *4 25 

Textile Color Mixing Svo, *4 25 

Paulding, C. P. Condensation of Steam in Covered and Bare 

Pipes Svo, *2 00 

Transmission of Heat Through Cold-storage Insulation 

i2mo, *i 00 

Payne, D. W. Founders' Manual Svo, *4 00 

Peckham, S. F. Solid Bitumens Svo, *5 00 

Peddie, R. A. Engineering and Metallurgical Books. . . . i2mo, *i 50 

Peirce, B. System of Analytic Mechanics 4to, 10 00 

Linnear Associative Algebra 4to, 3 00 

Pendred, V. The Railway Locomotive. (Westminster Series.) 

Svo, *2 00 

Perkin, F. M. Practical Method of Inorganic Chemistry . . i2mo, *i 00 

and Jaggers, E. M. Elementary Chemistry i2mo, *i 00 

Perrin, J. Atoms Svo, *2 50 

Perrine, F. A. C. Conductors for Electrical Distribution . . . Svo, *3 50 



D. VAN NOSTRAND COMPANY'S SHORT-TITLE CATALOG 33 

Petit, G. White Lead and Zinc White Paints 8vo, *2 50 

Petit, R. How to Build an Aeroplane. Trans, by T. O'B. 

Hubbard, and J. H. Ledeboer 8vo, *i 50 

Pettit, Lieut. J. S. Graphic Processes. (Science Series No. 76.) 

i6mo, o 50 
Philbrick, P. H. Beams and Girders. (Science Series No. 88.) 

i6mo, 

Phillips, J. Gold Assaying 8vo, *3 75 

Dangerous Goods 8vo, 3 50 

Phin, J. Seven Follies of Science i2mo, *i 25 

Pickworth, C. N. The Indicator Handbook. Two Volumes 

i2mo, each, i 50 

Logarithms for Beginners i2mo, boards, o 50 

The Slide Rule i2mo, j ^o 

Pilcher, R, B., and Butler-Jones, F. What Industry Owes 

to Chemical Science i2mo, i 50 

Plattner's Manual of Blowpipe Analysis. Eighth Edition, re- 
vised. Trans, by H. B. Cornwall 8vo, *4 00 

Plympton, G.W. The Aneroid Barometer. (Science Series.). i6mo, o 50 

How to become an Engineer. (Science Series No. 100.) 

i6mo, o 50 

Van Nostrand's Table Book. (Science Series No. 104). 

i6mo, o 50 
Pochet, M. L. Steam Injectors. Translated from the French. 

(Science Series No. 29.) i6mo, o 50 

Pocket Logarithms to Four Places. (Science Series.) i6mo, o 50 

leather, i 00 

PoUeyn, F. Dressings and Finishings for Textile Fabrics . 8vo, *4 25 

Pope, F. G. Organic Chemistry lamo, *2 50 

Pope, F. L. Modern Practice of the Electric Telegraph.. . 8vo, i 50 

Popplewell, W. C. Prevention of Smoke 8vo, *4 25 

Strength of Materials Bvo, *2 150 

Porritt, B. D. The Chemistry of Rubber. (Chemical Mono- 
graphs, No. 3.) i2mo, *i 00 

Porter, J. R. Helicopter Flying Machine i2mo, *i 50 

Potts, H. E. Chemistry of the Rubber Industry. (Outlines of 

Industrial Chemistry.) 8vo, *2 50 

Practical Compounding of Oils, Tallow and Grease 8vo, *4 25 



34 D. VAN NOSTRAND COMPANY S SHORT-TITLE CATALOG 

Pratt, K. Boiler Draught i2mo, *i 25 

High Speed Steam Engines. 8vo, *2 00 

Pray, T., Jr. Twenty Years with the Indicator 8vo, 2 50 

Steam Tables and Engine Constant 8vo, 2 00 

Prelini, C. Earth and Rock Excavation 8vo, *3 00 

Graphical Determination of Earth Slopes Svo, *2 00 

Tunneling 8vo, *3 00 

Dredging. A Practical Treatise Svo, *3 00 

Prescott, A. B. Organic Analysis Svo, 5 00 

and Johnson, 0. C. Qualitative Chemical Analysis . Svo, *3 50 

and Sullivan, E. C. First Book in Qualitative Chemistry 

i2mo, *i 50 

Prideaux, E. B. R. Problems in Physical Chemistry Svo, *2 00 

Theory and Usib of Indicators Svo, 5 00 

Prince, G. T. Flow of Water i2mo, *2 00 

Pullen, W. V/. F. Application of Graphic Methods to the Design 

of Structures i2mo, *2 50 

Injectors: Theory, Construction and Working. .. .i2mo, *2 00 

Indicator Diagrams Svo, *2 50 

Engine Testing Svo, *5 50 

Putsch, A. Gas and Coal-dust Firing Svo, *3 00 

Pynchon, T. R. Introduction to Chemical Physics Svo, 3 00 

Rafter, G. W. Mechanics of Ventilation. (Science Series No. 

33-) i6mo, o 50 

Potable Water. (Science Series No. 103.) i6mo, o 50 

Treatment of Septic Sewage, (Science Series No. 118.) 

i6mo, o 50 

and Baker, M. N. Sewage Disposal in the United States 

4to, *6 00 

Raikes, H. P. Sewage Disposal Works Svo, *4 00 

Randau, P. Enamels and Enamelling Svo, *7 25 

Rankine, W. J. M. Applied Mechanics Svo, 5 00 

Civil Engineering. Svo, 6 50 

Machinery and Millwork Svo, 5 00 

The Steam-engine and Other Prime Movers Svo, 5 00 

and Bamber, E. F. A Mechanical Textbook Svo, 3 50 



D. VAN NOSTRAND COMPANY'S SHORT-TITLE CATALOG 35 

Eansome, W. R. Freshman Mathematics i2mo, *i 35 

Raphael, F. C. Localization of Faults in Electric Light and 

Power Mains 8vo, *3 50 

Rasch, E. Electric Arc Phenomena. Trans, by K. Tornberg. 

8vo, *2 00 

Rathbone, R. L. B. Simple Jewellery 8 vo, *2 00 

Rateau, A. Flow of Steam through Nozzles and Orifices. 

Trans, by H. B. Brydon 8vo, *i 5c 

Rausenberger, F. The Theory of the Recoil of Guns..,.8vo, *5 00 
Rautenstrauch, W. Notes on the Elements of Machine Design, 

Svo, boards, *i 50 
Rautenstrauch, W., and Williams, J. T. Machine Drafting and 
Empirical Design. 

Part 1. Machine Drafting Svo, *i 25 

Part II. Empirical Design {In Preparation.) 

Raymond, E. B, Alternating Current Engineering i2mo, *2 50 

Rayner, H. Silk Throwing and Waste Silk Spinning. . .Svo, 
Recipes for the Color, Paint, Varnish, Oil, Soap and Drysaltery 

Trades Svo, *6 50 

Recipes for Flint Glass Making i2mo, *5 25 

Redfern, J. B., and Savin, J. Bells, Telephones. (Installa- 
tion Manuals Series.) . i6mo, *o 50 

Redgrove, H. S. Experimental Mensuration .i2mo, *i 25 

Redwood, B. Petroleum. (Science Series Ino. 92.) . . . .i6mo, o 50 

Reed, S. Turbines Applied to Marine Propulsion *5 00 

Reed's Engineers' Handbook Svo, *9 00 

Key to the Nineteenth Edition of Reed's Engineers* . 

Handbook Svo, *4 00 

Useful Hints to Sea-going Engineers i2mo, 3 00 

Reid, E. E. Introduction to Research in Organic Chemistry. 

{In Press.) 
Reid, H. A. Concrete and Reinforced Concrete Construction, 

Svo, *5 00 
Reinhardt, C. W. Lettering for Draftsmen, Engineers, and 

Students oblong 4to, boards, i 00 

■ The Technic of Mechanical Drafting, .oblong 4to, boards, *i 00 



36 D. VAN NOSTRAND COMPANY'S SHORT-TITLE CATALOG 

Reiser, F. Hardening and Tempering of Steel. Trans, by A. 

Morris and H. Robson i2mo, *3 00 

Reiser, N. Faults in the Manufacture of Woolen Goods. Trans. 

by A. Morris and H. Robson 8vo, 

< Spinning and Weaving Calculations 8vo, 

Renwick, W. G. Marble and Marble Working 8vo, 

Reuleaux, F. The Constructor. Trans, by H. H. Suplee. .4to, 

Reuterdahl, A. Theory and Design of Reinforced Concrete 

Arches 8vo, 

Rey, J. Range of Electric Searchlight Projectors Svo, 

Reynolds, 0., and Idell, F. E. Triple Expansion Engines. 

(Science Series No. 99.) i6mo, 

Rhead, G. F. Simple Structural Woodwork i2mo, 

Rhodes, H. J. Art of Lithography Svo, 

Rice, J. M., and Johnson, W. W. A New Method of Obtaining 

the Differential of Functions lamo, 

Richards, W. A. Forging of Iron and Steel lamo, 

Richards, W. A., and North, H. B. Manual of Cement Testing, 

i2mo, 

Richardson, J. The Modern Steam Engine Svo, 

Richardson, S. S. Magnetism and Electricity i2mo, 

Rideal, S. Glue and Glue Testing Svo, 

Riesenberg, F. The Men on Deck i2mo, 

Rimmer, E. J. Boiler Explosions, Collapses and Mishaps. Svo, 

Rings, F. Concrete in Theory and Practice i2mo, 

Reinforced Concrete Bridges 4to, 

Ripper, W. Course of Instruction in Machine Drawing., folio, 
Roberts, F. C. Figure of the Earth. (Science Series No. 79.) 

i6mo, o 50 
Roberts, J., Jr. Laboratory Work in Electrical Engineering 

Svo, *2 00 

Robertson, L. S. Water-tube Boilers Svo, 2 oa 

Robinson, J. B. Architectural Composition Svo, *2 50 

Robinson, S. W. Practical Treatise on the Teeth of Wheels. 

(Science Series No. 24.) i6mo, o 50 

Railroad Economics. (Science Series No. 59.) . . . . i6mo, o 50 

Wrought Iron Bridge Members. (Science Series No. 

60.) i6mo, o 50 



-^3 


00 


^6 


25 


5 


OQ 


*4 


GO 


*2 


GO 


''a 


50 





50 


*i 


25 


6 


50 





50 


I 


50 


*i 


50 


*3 


50 


*2 


GO 


*6 


50 


3 


GO 


*i 


75 


*2 


50 


*5 


GO 


*6 


GO 



D. VAN NOSTRAND COMPANY'S SHORT-TITLE CATALOG 37 

Robson, J. H. Machine Drawing and Sketching 8vo, 2 00 

Roebling, J. A. Long and Short Span Railway Bridges. . folio, 25 00 

Rogers, A. A Laboratory Guide of Industrial Chemistry . 8vo, 2 00 

Elements of Industrial Chemistry i2mo, 3 00 

Manual of Industrial Chemistry 8vo, *5 00 

Rogers, F. Magnetism of Iron Vessels. (Science Series No. 30.) 

r6mo, o 50 
Rohland, P. Colloidal and its Crystalloidal State of Mattel. 

Trans, by W. J. Britland and H. E. Potts . . i2mo, *i 25 

Rollinson, C. Alphabets obloii|^ i2ino, *i 00 

Rose, J. The Pattern-makers' Assistant 8vo, 2 50 

Key to Engines and Engine-running i2mo, 2 50 

Rose, T. K. The Precious Metals. (Westminster Series.) . .8vo, *2 00 
Rosenhain, W. Glass Manufacture. (Westminster Series.) . .8vo, *2 00 
Physical Metallurgy, An Introduction to. (Metallurgy 

Series.) Svo, *3 50 

Roth, W. A. Physical Chemistry Svo, *2 00 

Rowan, F. J. Practical Physics of the Modern Steam-boiler. Svo, *3 00 
and Idell, F. E. Boiler Incrustation and Corrosion. 

(Science Series IVo. 27.) i6mo, o 50 

Roxburgh, W. General Foundry Practice. (Westminster 

Series.) Svo, *2 00 

Ruhmer, E. Wireless Telephony. Trans, by J. Erskine- 

Murray Svo, *4 50 

Russell, A. Theory of Electric Cables and Networks Svo, *3 00 

Rutley, F. Elements of Mineralogy i2mo, *i 25 

Sandeman, E. A. The Manufacture of Earthenware. .i2mo, 3 50 

Sanford, P. G. Nitro-explosives Svo, *4 00 

Saunders, C. H. Handbook of Practical Mechanics i6mo, i 00 

leather, i 25 

Sayers, H. M. Brakes for Tram Cars Svo, *i 25 

Scheele, C. W. Chemical Essays Svo, *2 00 

Scheithauer, W. Shale Oils and Tars Svo, *5 00 

Scherer, R. Casein. Trans, by C. Salter Svo, *4 25 



*6 


oo 


*2 


25 


*I 


75 


*3 


OQ 


I 


50 


*3 


00 


*4 


.SO 


2 


50 


*6 


00 



38 D. VAN NOSTRAND COMPANY'S SHORT-TITLE CATALOG 

Schidrowitz, P. Rubber, Its Production and Industrial Uses, 

8vo, 

Schindler, K. Iron and Steel Construction Works lamo, 

Schmall, C. N. First Course in Analytic Geometry, Plane and 

Solid i2mo, half leather, 

Schmeer, L. Flow of Water 8vo, 

Schumann, F. A Manual of Heating and Ventilation. 

i2mo, leather, 

Schwartz, E. H. L. Causal Geology 8vo, 

Schweizer, V. Distillation of Resins 8vo, 

Scott, W. W. Qualitative Analysis. A Laboratory Manual, 
New Edition 

Standard Methods of Chemical Analysis 8vo, 

Scribner, J. M, Engineers' and Mechanics' Companion. 

i6mo, leather, i 50 
Scudder, H. Electrical Conductivity and Ionization Constants 

of Organic Compounds Svo, 

Searle, A, B. Mcdern Brickmaking Svo, 

Cement, Concrete and Bricks Svo, 

Searle, G. M. " Sumners' Method." Condensed and Improved. 

(Science Series No. 124.) i6mo, 

Seaton, A. E. Manual of Marine Engineering .Svo, 

Seaton, A. E., and Rounthwaite, H. M. Pocket-book of Marine 

Engineering i6mo, leather, 5 qo 

Seeligmann, T., Torrilhon, G. L., and Falconnet, H. India 
Rubber and Gutta Percha. Tra.is. by J. G. Mcintosh 

Svo, *7 25 
Seidell, A. Solubilities of Inorganic and Organic Substances, 

Svo, *3 00 

Seligman, R. Aluminum. (Metallurgy Series) {In Press.} 

Sellew, W. H. Steel Rails 4to, *io 00 

Railway Maintenance Engineering i2mo, *2 50 

Senter, G. Outlines of Physical Chemistry. i2mo, -''2 00 

Textbook of Inorganic Chemistry i2mo, *3 00 

Sever, G.F. Electric Engineering Experiments ... 8vo, boards, *i 00 
and Townsend, F. Laboratory and Factory Tests in Elec- 
trical Engineering Svo, *2 50 



*3 


00 


*7 


25 


-^6 


5f> 





50 


8 


00 



D, VAN NOSTRAND COMPANY'S SHORT-TITLE CATALOG 39 

Sewall, C. H. Wireless Telegraphy 8vo, *2 oo 

Lessons in Telegraphy i2mo, *i oo 

Sewell, T. The Construction of Dynamos 8vo, *3 oo 

Sexton, A. H. Fuel and Refractory Materials i2mo, *2 50 

Chemistry of the Materials of Engineering . . .. i2mo, *2 50 

Alloys (Non- Ferrous) 8vo, *3 00 

and Primrose, J. S. G. The Metallurgy of Iron and Steel, 

8vo, *6 50 

Seymour, A. Modern Printing Inks 8vo, *3 00 

Shaw, Henry S. H, Mechanical Integrators. (Science Series 

No. 83.) i6mo, o 50 

Shaw, S. History of the Staffordshire Potteries 8vo, 3 00 

Chemistry of Compounds Used in Porcelain Manufacture, 

8vo, *6 oo 

Shaw, T. R. Driving of Machine Tools i2mo, *2 50 

Precision Grinding Machines lamo, 5 50 

Shaw, W. N. Forecasting Weather 8vo, *3 50 

Sheldon, S., and Hausmann, E. Direct Current Machines. i2mo, *2 50 

Alternating-current Machines i2mo, *2 50 

Electric Traction and Transmission Engineering. .i2mo, *2 50 

Physical Laboratory Experiments 8vo, *i 25 

Shields, J. E. Note on Engineering Construction i2mo, i 50 

Shreve, S. H. Strength of Bridges and Roofs 8vo, 3 50 

Shunk, W. F. The Field Engineer i2mo, fabrikoid, 2 50 

Simmons, W. H., and Appleton, H. A. Handbook of Soap 

Manufacture 8vo, *5 00 

Simmons, W. H., and Mitchell, C. A. Edible Fats and Oils, 

8vo, *4 50 

Simpson, G. The Naval Constructor i2mo, fabrikoid, *5 00 

Simpson, W. Foundations 8vo (hi Press.} 

Sinclair, A. Development of the Locomotive Engine. 

8vo, half leather, 5 00 
Sindall, R. W. Manufacture of Paper. (Westminster Series.) 

8vo, *2 00 

and Bacon, W. N. The Testing of Wood Pulp 8vo, *2 50 

Sloane, T. O'C. Elementary Electrical Calculations ..... i2mo, *2 00 
Smallwood, J. C. Mechanical Laboratory Methods. (Van 

Nostrand's Textbooks.) i2mo, fabrikoid, *3 00 

Smith, C. A. M. Handbook of Testing, MATERIALS. .8vo, *2 50 

and Warren, A. G. New Steam Tables 8vo, *i 25 

Smith, C. F. Practical Alternating Currents and Testing. 8 vo, *3 50 
Practical Testing of Dynamos and Motors Svo, *3 00. 



40 D. VAN NOSTRAND COMPANY S SHORT-TITLE CATALOG 

Smith, F. A. Railway Curves i2mo, *i oo 

Standard Turnouts on American Railroads i2mo, *i oo 

. Maintenance of Way Standards i2mo, *i 50 

Smith, F. E. Handbook of General Instruction for Mechanics. 

i2mo, I 50 
Smith, G. C. Trinitrotoluenes and Mono- and Dinitroto- 

luenes, Their Manufacture and Properties i2mo, 2 00 

Smith, H. G. Minerals and the Microscope i2mo, *i 25 

Smith, J. C. Manufacture of Paint 8vo, *3 50 

Smith, R. H. Principles of Machine Work i2mo, 

Advanced Machine Work i2mo, *3 00 

Smith, W. Chemistry of Hat Manufacturing i2mo, *4 50 

Snell, A. T. Electric Motive Power 8vo, *4 00 

Snow, W. G. Pocketbook of Steam Heating and Ventilation, . 

{In Press.) 
Snow, W. G., and Nolan, T. Ventilation of Buildings. (Science 

Series No. 5.) i6mo, o 50 

Soddy, F. Radioactivity 8vo, *3 00 

Solomnn, M. Electric Lamps. (Westminster Series.) 8vo, *2 00 

Somerscales, A. N. Mechanics for Marine Engineers. .i2mo, *2 00 

Mechanical and Marine Engineering Science 8vo, *5 00 

Sothern, J. W. The Marine Steam Turbine 8vo, *i5 00 

■ Verbal Notes and Sketches for Marine Engineers .... Svo, *9 00 

Sothern, J. W., and Sothern, R. M. Elementary Mathematics 

for Marine Engineers i2mo, *i 50 

Simple Problems in Marine Engineering Design. .i2mo, 

Southcombe, J. E. Chemistry of the Oil Industries. (Out- 
lines of Industrial Chemistry) Svo, *3 00 

Soxhlet, D. H. Dyeing and Staining Marble. Trans, by A. 

Morris and H. Robson Svo, *3 00 

Spangenburg, L. Fatigue of Metals. Translated by S. H. 

Shreve. (Science Series No. 23.) i6mo, o 50 

Specht, G. J., Hardy, A. S., McMaster, J. B., and Walling. Topo- 
graphical Surveying. (Science Series No. 72.). . i6mo, o 50 

Spencer, A. S. Design of Steel-Framed Sheds Svo, *3 50 

Speyers, C. L. Text-book of Physical Chemistry Svo, *i 50 

Spiegel, L. Chemical Constitution and Physiological Action. 

(Trans, by C. LuedeHng and A. C. Poylston.) .i2mo, *i 25 



X) VAN NOSTRAND COMPANY S SHORT-TITLE CATALOG 4I 

Sprague, E. H. Elementary Mathematics for Engineers i2mo, 2 50 

— — Elements of Graphic Statics 8vo, 2 50 

Sprague, E. H. Hydraulics i2mo, *3 00 

Stability of Arches i2mo, 2 50 

Stability of Masonry i2mo, *2 50 

Strength of Structural Elements i2mo, 2 00 

,Stahl, A. W. Transmission of Power. (Science Series No. 28.) 

i6mo, 

-■ and Woods, A. T. Elementary Mechanism i2mo, *2 00 

Staley, C, and Pierson, G. S. The Separate System of 

Sewerage Svo, *3 00 

Standage, H. C. Leatherworkers' Manual Svo, *4 50 

Sealing Waxes, Wafers, and Other Adhesives Svo, *2 50 

Agglutinants of All Kinds for All Purposes i2mo, *^ go 

Stanley, H. Practical Applied Physics (/;/ Press.) 

Stansbie, J. H. Iron and Steel. (Westminster Series.) . .Svo, *2 00 

Steadman, F. M. Unit Photography i2mo, *2 00 

Stecher, G. E. Cork. Its Origin and Industrial Uses..i2mo, i 00 
Steinman, D, B. Suspension Bridges and Cantilevers. 

(Science Series No. 127.) o 50 

Melan's Steel Arches and Suspension Bridges Svo, *3 oc 

Stevens, E. J. Field Telephones and Telegraphs for Army 

Use i2m'0, I 20 

Stevens, H. P. Paper Mill Chemist i6mo, *4 25 

Stevens, J. S. Theory of Measurements i2mo, *i 25 

Stevenson, J. L, Blast-Furnace Calculations. .i2mo, leather, *2 00 

Stewart, G. Modern Steam Traps i2mo, *i 75 

Stiles, A. Tables for Field Engineers i2mo, i 00 

Stodola, A. Steam Turbines. Trans by L. C. Loewenstein.Svo, *5 00 

Stone, H. The Timbers of Commerce Svo, 3 50 

Stopes, M. Ancient Plants Svo, *2 00 

The Study of Plant Life Svo, *2 00 

Sudborough, J. J., and James, T. C. Practical Organic Chem- 
istry i2mo, *2 00 

Suf fling, E. R. Treatise on the Art of Glass Painting. .Svo, *4 25 
Sullivan, T. V., and Underwood, N. Testing and Valuation 

of Building and Engineering Materials (In Press.) 

Sur, F. J. S. Oil Prospecting and Extracting Svo, *i 00 

Svenson, C. L. Handbook of Piping 4 00 

Essentials of Drafting Svo, i 50 

^wan, K. Patents, Designs and Trade Marks. (Westminster 

Series.) Svo, *2 00 



42 D. VAN NOSTRAND COMPANY S SHORT-TITLE CATALOG 

Swinburne, J., Wordingham, C. H., and Martin, T. C. Electric 

Currents. (Science Series No. 109.) i6mo, o 50- 

Swoope, C. W. Lessons in Practical Electricity i2mo, *2 00 

Tailfer, L. Bleaching Linen and Cotton Yarn and Fabrics.Svo, *8 50 
Tate, J. S. Surcharged and Different Forms of Retaining- 

walls. (Science Series No. 7.) . i6mo, o 50 

Taylor, F. N. Small Water Supplies i2mo, *2 50 

Masonry in Civil Engineering 8vo, ■'2 50 

Taylor, T. U. Surveyor's Handbook i2mo, leather, *2 00 

Backbone of Perspective i2mo, *i 00 

Taylor, W. P. Practical Cement Testing 8vo, *3 00 

Templeton, W. Practical Mechanic's Workshop Companion, 

i2mo, morocco, 2 00 
Tenney, E. H. Test Methods for Steam Power Plants. 

(Van Nostrand's Textbooks.) lamo, *2 50- 

Terry, H. L. India Rubber and its Manufacture. (West- 
minster Series.) 8vo, *2 oo- 

Thayer, H. R. Structural Design 8vo, 

Vol. I, Elements of Structural Design *2 00 

Vol. IL Design of Simple Structures *4 00 

Vol. III. Design of Advanced Structures {In Preparation.) 

Foundations and Masonry {In Preparation.) 

Thiess, J. B., and Joy, G. A. Toll Telephone Practice. .8vo, *3 50 

Thom, C; and Jones, W. H. Telegraphic Connections, 

Thomas, C W, Paper-makers' Handbook. {In Press.) 

oblong i2mo, i 50 

Thomas, J. B Strength of Ships 8vo, 300 

Thomas, Robt. G. Applied Calculus i2mo (In Press.) 

Thompson, A. B Oil Fields of Russia 4to, *7 5© 

Oil Field Development and Petroleum Mining 8vo, *7 5° 

Thompson, S P Dynamo Electric Machines. ''Science 

Series No. 75.) i6mo, o 50 

Thompson, W P. Handbook of Patent Law of All Countries, 

i6mo, I 50 

Thomson, G Modern Sanitary Engineering lamo, *3 00 

Thomson, G. S. Milk and Cream Testing lamo, *2 25 

- — Modern Sanitary Engineering, House Drainage, etc. .8vo, *3 oc- 



D. VAN NOSTRAND COMPANY S SHORT-TITLE CATALOG 43, 

Thornley, T. Cotton Combing Machines 8vo, *3 00 

Cotton Waste 8vo, *4 50 

Cotton Spinning 8vo, 

First Year '^2 00 

Second Year *4 25 

Third Year *3 50 

Thurso, J. W. Modern Turbine Practice Svo, *4 00 

Tidy, C. Meymott. Treatment of Sewage. (Science Series 

No. 94.) , i6mo, o 50 

Tillmans, J. Water Purification and Sewage Disposal. Trans. 

by Hugh S. Taylor Svo, *2 00 

Tinney, W. H. Gold-mining Machinery Svo, *3 00 

Titherley, A. W. Laboratory Course of Organic Chemistry.Svo, *2 00 

Tizard, H. T. Indicators {In Press.) 

Toch, M. Chemistry and Technology of Paints Svo, *4 00 

Materials for Permanent Painting i2mo, *2 00 

Tod, J., and McGibbon, W. C. Marine Engineers' Board of 

Trade Examinations Svo, *2 00 

Todd, J., and Whall, W. B. Practical Seamanship Svo, S 00 

Tonge, J. Coal. (Westminster Series.) Svo, *2 00 

Townsend, F. Alternating Current Engineering. .Svo, boards, *o 75 

Townsend, J. Ionization of Gases by Collision Svo, *i 25. 

Transactions of the American Institute of Chemical Engineers. 
Eight volumes now ready. Vols. I. to IX., 1908-1916, 

Svo, each, *6 00 
Vol. X. Un Press.) 

Traverse Tables. (Science Series No. 115.) i6mo, o 50 

mor., I 00 

Treiber, E. Foundry Machinery. Trans, by C. Salter. . i2mo, *i 50 
Trinks, W., and Housum, C. Shaft Governors. (Science 

Series No. 122.) i6mo, o 50 

Trowbridge, D. C. Handbook for Engineering Draughtsmen. 

{In Press.) 
Trowbridge, W. P. Turbine Wheels. (Science Series No. 44.) 

i6mo, o 50 

Tucker, J. H. A Manual of Sugar Analysis Svo, 3 50 

Tunner, P. A, Treatise on Roll-turning. Trans, by J. B. 

Pearse Svo text and folio atlas, 10 00 



44 D. VAN NOSTRAND COMPANY S SHORT-TITLE CATALOG 

Turnbull, Jr., J., and Robinson, S. W. A Treatise on the ' 
Compound Steam-engine. (Science Series No. 8.) 

i6mo. 

Turner, H, Worsted Spinners' Handbook i2mo, *3 50 

Turril], S. M. Elementary Course in Perspective i2mo, *i 25 

Twyford, H. B. Purchasing 8vo, *3 00 

Storing, Its Economic Aspects and Proper Methods. .8vo, 3 50 

Tyrrell, H. G. Design and Construction of Mill Buildings. 8vo, *4 00 

Concrete Bridges and Culverts i6mo, leather, *3 00 

Artistic Bridge Design Bvo, *3 00 

Underhill, C. R. Solenoids, Electromagnets and Electromag- 
netic Windings i2mo, *2 00 

Underwood, N., and Sullivan, T. V. Chemistry and Tech- 
nology of Printing Inks 8vo, 

Urquhart, J. W. Electro-plating i2mo, 

Electrotyping i2mo, 

TJsborne, P. 0. G. Design of Simple Steel Bridges ...... 8vo, 

Vacher, F, Food Inspector's Handbook i2mo, 

Van 'Nostra nd'is Chemical Annual. Fourth Issue 1913. 

fiabrikoid. i2m0, 

Year Book of Mechanical Engineering Data (In 

Van Wagenen, T. F. Manual of Hydraulic Mining i6mo, i 00 

Vega, Baron, Von. Logarithmic Tables 8vo, 2 50 

Vincent, C. Ammonia and its Compounds . Trans, by M. J. 

Salter 8vo, *3 00 

Volk, C. Haulage and Winding Appliances 8vo, *4 00 

Von Georgievics, G. Chemical Technology of Textile Fibres. 

Trans, by C Salter 8vo, 

• Chemistr:^ of Dyestuffs. Trans, by C. Salter 8vo, *4 50 

Vose, G. L. Graphic Method for Solving Certain Questions in 

Arithmetic and Algebra. (Science Series No. 16.) 

i6mo, o 50 

Vosmaer, A. Ozone 8vo, *2 59 

Wabner, R. Ventilation in Mines. Trans, by C. Salter. .8vo, *6 50 
Wade, E. J. Secondary Batteries 8vo, *4 o 



-3 


00 


2 


00 


2 


00 


*4 


00 


*3 


00 


*3 


00 


Press.) 



D. VAN NOSTRAND COMPANY'S SHORT-TITLE CATALOG 45 

Wadmore, J. M. Elementary Chemical Theory i2mo, *i 50 

Wagner, E. Preserving Fruits, Vegetables, and Meat..i2mo, *3 00 
Wagner, J. B. A Treatise on the Natural and Artificial 

Processes of Wood Seasoning 8vo, 300 

Waldram, P. J. Principles of Structural Mechanics. .. i2mo, *3 00 
Walkter, F. Dynamo Building. (Science Series No. 98.) 

i6mo, o 50 

Walker, J. Organic Chemistry for Students of Medicine. 8vo, *3 00 

Walker, S. F. Steam Boilers, Engines and Turbines. .. .8vo, 3 00 

Refrigeration, Heating and Ventilation on Shipboard, 

i2mo, *2 00 
Electricity in Mining Bvo, '% 50 

Wallis-Tayler, A. J. Bearings and Lubrication 8vo, *i 50 

Aerial or Wire Ropeways 8vo, *3 00 

Preservation of Wood 8vo, 4 00 

Refrigeration, Cold Storage and Ice Making 8vo, 5 50 

Sugar Machinery i2mo, *2 50 

Walsh, J, J. Chemistry and Physics of Mining and Mine 

Ventilation i2mo, *2 00 

Wanklyn, J. A. Water Analysis i2mo, 2 00 

Wansbrough, W. D. The A B C of the Differential Calculus, 

i2mo, *2 50 

Slide Valves i2mo, *2 00 

Waring, Jr., G. E. Sanitary Conditions. (Science Series 

No. 31.) i6mo, o 50 

Sewerage and Land Drainage *6 00 

Modern Methods of Sewage Disposal i2mo, 2 00 

How to Drain a House i2mo, • i 25 

Warnes,^ A. R. Coal Tar Distillation 8vo, *5 00 

Warren, F. D. Handbook on Reinforced Concrete i2mo, '2 50 

Watkins, A. Photography. (Westminster Series.) 8vo, *2 00 

Watson, E. P. Small Engines and Boilers i2mo, i 25 

Watt, A. Electro-plating and Electro-refining of Metals. 8vo, *4 50 

Electro-metallurgy i2mo, i 00 

The Art of Soap-making 8 vo, 4 00 

Leather Manufacture Bvo, *6 00 

Paper Making 8vo, 3 75 

Webb, H. L. Guide to the Testing of Insulated Wires and 

Cables i2mo, i 00 

Webber, W, H. Y. Town Gas. (Westminster Series.). ..8vo, *2 00 
Wegmann, E. Conveyance and Distribution of Water for 

Water Supply 8vo, 5 00 



*6 


00 


*7 


50 


*3 


75 


*2 


50 


3 


00 


*"! 


50 


*2 


25 



46 D. VAN NOSTRAND COMPANY^S SHORT-TITLE CATALOG 

Weisbach, J. A Manual of Theoretical Mechanics 8vo, 

sheep, 

and Herrmann, G. Mechanics of Air Machinery. .. .8vo, 

Wells, M. B. Steel Bridge Designing 8vo, 

Wells, Robt. Ornamental Confectionery lamo, 

Weston, E. B. Loss of Head Due to Friction of Water in Pipes, 

i2mo, 

Wheatley, 0. Ornamental Cement Work 8vo, 

Whipple, S. An Elementary and Practical Treatise on Bridge 

Building 8vo, 3 00 

White, C. H. Methods in Metallurgical Analysis. (Van 

Nostrand's Textbooks.) lamo, 

White, G. F. Qualitative Chemical Analysis lanio. 

White, G. T. Toothed Gearing i2mo, 

White, H. J. Oil Tank Steamers i2mo, paper, 

Whitelaw, John. Surveying 8vo, 

Widmer, E. J. Observation Balloons : i2mo, 

Wilcox, R. M. Cantilever Bridges. (Science Series No. 25.) 

i6mo, 

Wilda, H. Steam Turbines. Trans, by C. Salter i2mo, 

Cranes and Hoists. Trans, by Chas. Salter i2mo, 

Vnikinson, H. D. Subm^arine Cable Laying and Repairing . 8vo, 

Williamson, J. Surveying 8vo, 

Williamson, R. S. On the Use of the Barometer .4to, 

Practical Tables in Meteorology and Hypsometry. .4to, 

Wilson, F. J., and Heilbron, I. M. Chemical Theory and Cal- 
culations i2mo, 

Wilson, J. F. Essentials of Electrical Engineering 8vo, 

Wimperis, H. E. Internal Combustion Engine 8vo, 

Application of Power to Road Transport ,i2mo, 

Primer of Internal Combustion Engine i2mo, 

Winchell, N. H., and A. N. Elements of Optical Mineralogy .8 vo, 
Winslow, A. Stadia Surveying. (Science Series No. 77.) . i6mo, 
Wisser, Lieut. J. P. Explosive Materials. (Science Series No. 

70.) i6mo, o 50 

Wisser, Lieut. J. P. Modern Gun Cotton. (Science Series No. 

89. ) i6mo, o 50 

Wolff, C. E. Modern Locomotive Practice 8vo, *4 20 



2 


50 


*I 


25 


*2 


50 


I 


50 


4 


50 


3 


00 





50 


*2 


50 


=^2 


50 


*6 


00 


*3 


00 


15 


00 


2 


50 


*i 


00 


2 


50 


*3 


00 


*i 


50 


*i 


00 


*3 


50 





50 



D. VAN NOSTRAND COMPANY^S SHORT-TITLE CATALOG 4/ 

Wood, De V. Luminiferous Aether. (Science Series No. 85.) 

i6mo, o 50 
Wood, J. K. Chemistry of Dyeing. (Chemical Monographs 

No. 2.) i2mo, *i 00 

Worden, E. C. The Nitrocellulose Industry. Two vols..8vo, *io 00 

Technology of Cellulose Esters. In 10 vols 8vo. 

Vol. VIII. Cellulose Acetate *5 00 

Wren, H. Organometallic Compounds of Zinc and Magnesium. 

(Chemical Monographs No. i.) i2mo, *i 00 

Wright, A. C. Analysis of Oils and Allied Substances 8vo, *3 50 

. Simple Method for Testing Painter's Materials. .. .8vo, *3 00 

Wright, F. W. Design of a Condensing Plant i2mo, *i 50 

Wright, H. E. Handy Book for Brewers Svo, *6 00 

Wright, J. Testing, Fault Finding, etc. for Wiremen (Installa- 
tion Manuals Series) i6mo, *o 50 

Wright, T. W. Elements of Mechanics Svo, *2 50 

and Ilayf ord, J. F. Adjustment of Observations .... Svo, *3 00 

Wynne, W. E., and Spraragen, W. Handbook of Engineerinf 

Mathematics i2mo, leather, *2 00 

Yoder, J. H. and Wharen, G. B. Locomotive Valves and 

Valve Gears Svo, 3 00 

Young, J. E. Electrical Testing for Telegraph Engineers. . .Svo, *4 00 

Youngson, P. Slide Valve and Valve Gearing 4to, 3 00 

Zahner, R. Transmission of Power. (Science Series No. 40.) 

i6mo, 

2eidler, J., and Lustgarten, J. Electric Arc Lamps Svo, *2 00 

Zeuner, A. Technical Thermodynamics. Trans, by J. F. 

Klein. Two Volumes .Svo, *S 00 

Zimmer, G. F. Mechanical Handling and Storing of Materials, 

4to, *I2 50 

Mechanical Handling of Material and Its National Im- 
portance During and After the War 4to, 4 00 

Zipser, J. Textile Raw Materials. Trans, by C. Salter. .Svo, *6 25 

Zur Nedden, F. Engineering Workshop Machines and Proc- 
esses. Trans, by J. A, Davenport Svo, *2 00 



D.Van nostrand Company 

are prepared to supply, either from 

their complete stock or at 

short notice, 

Any Technical or 

Scientific Book 

In addition to publishing a very large 
and varied number of Scientific and 
Engineering Books, D. Van Nostrand 
Company have on hand the largest 
assortment in the United States of such 
books issued by American and foreign 
publishers. 



All inquiries are cheerfully and care- 
fully answered and complete catalogs 
sent free on request. 



25 Park Place - - - New York 

10 M. Feb. 8—1919 ~~ ' 

ij- U ij rj 







/» I^^W^'» 'T- *^ %9 9l* •* •» *!^^^W • 






*' 



'^^**'T:t«' ^-v 



I* .""•."*©. ** A^ .."•♦ "^^ 

















.«»^, 






^«. 









'bV 






V -^^o* 



bV 






^°-n^. 










>^ 6-A-* '^c 



o, ♦/T7i» A 



<> 'o 




"v-^^ 



L<^* .^'** ^ 



V^ . • • 



























.•«-' 






0^ ©•••♦. 'o 



V'^*.*<^..,. 






i?^^^ ..! 






lO- 



r. %.A^ : 




* <^ \ 'M 

• j^^J^tw^*^^ O 



♦ rxO 






•^'o -^^^ .^* ^ 










i**\-^^vV ,c<^.-^^'>o .4**\v;^..\. 

40, 




^'•• 



""^ 

.^^ 









y 



V • 









^♦^ >^ 






